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A Parabolic Temperature Profile Model for Heating of Droplets
Abstract
A model for convective heating of droplets, which takes into account their finite thermal conductivity, is suggested. This model is based on the assumption of the parabolic temperature profile in the droplets. A rigorous numerical solution, without restrictions on temperature profiles inside droplets, is compared with predictions of the parabolic temperature profile and isothermal models. The comparison shows the applicability of the parabolic approximation to modelling of the heating of fuel droplets in realistic diesel engines. The simplicity of the model makes it particularly convenient for implementation into CFD codes.
... Assuming that initially T (R) is constant, one can see that, except at the very start of heating, the shape of the curve T (R) looks very close to a parabola. This allows us to approximate T (R) as [51] T ...
... For most practical applications, we are primarily interested in the values of T s , which determine the rate of evaporation and break-up of droplets. The model based on Expression (2.43) was called the parabolic temperature profile model [51]. ...
... This model shows good accuracy at long times, but can differ considerably from the results based on the numerical solution to Eq. (2.1) for short times [51]. The period of time when this deviation happens is usually short and can be ignored in most practical calculations. ...
The analysis of heating of non-evaporating spherical droplets is based on the assumption that they retain their forms, and the temperatures over the whole droplet surfaces are the same (although they can vary with time). The models for convective and radiative heating of droplets are considered separately. The following models are presented: coupled analytical solutions for the temperature of stationary droplets, separate solutions for gas and liquid phases in several limiting cases and heating of moving droplets (steady-state and transient cases). Simplified models for the heating of the liquid phase, based on the parabolic model and its modifications, are described. The radiative heating of droplets is described, using the following approaches: the most general Mie theory, analysis of integral absorption of radiation in droplets and geometric optics analysis. It is pointed out that the approach based on the integral absorption of thermal radiation inside droplets is applicable for the analysis of radiative heating of droplets in internal combustion engine conditions.
... Assuming that initially T (R) is constant, one can see that, except at the very start of heating, the shape of the curve T (R) looks very close to a parabola. This allows us to approximate T (R) as [51] T ...
... For most practical applications, we are primarily interested in the values of T s , which determine the rate of evaporation and break-up of droplets. The model based on Expression (2.43) was called the parabolic temperature profile model [51]. ...
... This model shows good accuracy at long times, but can differ considerably from the results based on the numerical solution to Eq. (2.1) for short times [51]. The period of time when this deviation happens is usually short and can be ignored in most practical calculations. ...
The models used for the analysis of mono-component droplet heating and evaporation in this chapter assume that vapour in the vicinity of the droplet surface is saturated. Hence, the rate of droplet evaporation is equal to the rate of vapour diffusion from its surface to ambient gas. These are known as the hydrodynamic models of droplet evaporation. The analysis starts with empirical correlations which are not directly linked with any evaporation model. Then classical hydrodynamic models of droplet evaporation are described. All of these models assume that the droplet’s radius remains constant during the time step but changes from one time step to another due to droplet thermal swelling and evaporation. Then the effects of droplet radii change during individual time steps on the heating process are investigated. Approaches to modelling heating and evaporation of spheroidal droplets are presented. Previously developed tools for modelling radiative heating of droplets are adapted to modelling the effects of support on droplet heating and evaporation. Comparisons with experimental results are described.
... Assuming that initially T (R) is constant, one can see that, except at the very start of heating, the shape of the curve T (R) looks very close to a parabola. This allows us to approximate T (R) as [51] T ...
... For most practical applications, we are primarily interested in the values of T s , which determine the rate of evaporation and break-up of droplets. The model based on Expression (2.43) was called the parabolic temperature profile model [51]. ...
... This model shows good accuracy at long times, but can differ considerably from the results based on the numerical solution to Eq. (2.1) for short times [51]. The period of time when this deviation happens is usually short and can be ignored in most practical calculations. ...
The analysis of heating of non-evaporating droplets is based on the assumptions that they retain their spherical form and that the temperature over their whole surface is the same (although it can vary with time). The models for convective and radiative heating of droplets are considered separately. The following cases of convective heating are considered: coupled analytical solutions for stationary droplets, separate solutions for gas and liquid phases in some limiting cases, and heating of moving droplets (steady-state and transient cases). The radiative heating of droplets is considered, using the following approaches: the most general Mie theory, analysis of integral absorption of radiation in droplets, and geometric optics analysis.
... By checking droplet temperature distribution, T p (r, t), one observes that, except at the very beginning of heating, the shape of the curve T p (r) looks close to a parabola [51]. Hence, the finite rate heat conduction process is taken into account by assuming that the temperature profile between the droplet surface and its center is a parabola [15]: ...
... where T cntr is the temperature at the droplet center (at r = 0) and T surf is the droplet surface temperature (at r = R p , with R p the droplet radius). If we generalize the derivation of [15] for evaporating droplets as done in [13], we can consider the volume averaged droplet temperature T p , defined as: ...
... After considering the boundary condition at the droplet surface as in [15], its time evolution then reads: ...
This paper presents a numerical modeling study of one ethanol spray flame from the Delft Spray-in-Hot-Coflow (DSHC) database, which has been used to study Moderate or Intense Low-oxygen Dilution (MILD) combustion of liquid fuels (Correia Rodrigues et al. Combust. Flame 162(3), 759-773, 2015). A "Lagrangian-Lagrangian" approach is adopted where both the joint velocity-scalar Probability Density Function (PDF) for the continuous phase and the joint PDF of droplet properties are modeled and solved. The evolution of the gas phase composition is described by a Flamelet Generated Manifold (FGM) and the interaction by exchange with the mean (IEM) micro-mixing model. Effects of finite conductivity on droplet heating and evaporation are accounted for. The inlet boundary conditions starting in the dilute spray region are obtained from the available experimental data together with the results of a calculation of the spray including the dense region using ANSYS Fluent 15. A method is developed to determine a good estimation for the initial droplet temperature. The inclusion of the "1/3" rule for droplet evaporation and dispersion models is shown to be very important. The current modeling approach is capable of accurately predicting main properties, including mean velocity, droplet mean diameter and number density. The gas temperature is under-predicted in the region where the enthalpy loss due to droplet evaporation is important. The flame structure analysis reveals the existence of two heat release regions, respectively having the characteristics of a premixed and a diffusion flame. The experimental and modeled temperature PDFs are compared, highlighting the capabilities and limitations of the proposed model.
... Two-temperature approach can be derived more rigorously if a particular functional form of internal temperature distribution is assumed. This is the idea of parabolic approximation when the radial temperature distribution is assumed to be second order polynomial as suggested by Dombrovsky and Sazhin [2003] (note, that this approximation was earlier used for modelling diffusion of lithium in spherical particles of active material, e.g. by Subramanian et al. [2001]). The droplet core temperature is formally defined as volume-averaged temperature which is coupled (via heat balance ODE) to droplet surface temperature and the heat flux received by the liquid. ...
... Parabolic approximation. Temperature distribution inside the droplet is approximated by the second order polynomial (for example, see Dombrovsky and Sazhin [2003], Subramanian et al. [2001]): ...
... This clearly demonstrates that parabolic approximation is invalid, since at l t the quasi-steady parabolic profile does not yet exists. To relax this controversy, it has been suggested by Dombrovsky and Sazhin [2003] to artificially smooth the transient dependence s Tt at small times by applying the following ad hoc relation: ...
Two simple and yet sufficiently accurate approaches to predict surface temperature of a vaporizing droplet (higher order polynomial approximation and the heat balance integral method) are proposed. Being computationally inexpensive these approaches are tested as candidates for use in high-resolution LES spray modelling. Robust and efficient numerical algorithm for solving inherently stiff equations of droplet heating and evaporation has been developed. Robustness and computational efficiency of the proposed algorithm is achieved by use of unconditionally stable strongly implicit integration scheme and appropriate adaptation of the time step. The above methodology has been implemented in CFD spray model included in the in-house Fire3D code. The spray model has been applied to replicate three essentially different experimental scenarios in which turbulent sprays of water, acetone, and diesel fuel were investigated. Reasonable agreement has been demonstrated for predicted and measured droplet sizes and velocities as well as for the spray tip penetration dynamics. New numerical algorithms used to calculate surface temperatures of evaporating droplets with non-uniform internal temperature did not incur observable increase of CPU time in turbulent spray simulations.
... Two-temperature approach can be derived more rigorously if a particular functional form of internal temperature distribution is assumed. This is the idea of parabolic approximation when the radial temperature distribution is assumed to be second order polynomial as suggested by Dombrovsky and Sazhin [2003] (note, that this approximation was earlier used for modelling diffusion of lithium in spherical particles of active material, e.g. by Subramanian et al. [2001]). The droplet core temperature is formally defined as volume-averaged temperature which is coupled (via heat balance ODE) to droplet surface temperature and the heat flux received by the liquid. ...
... Parabolic approximation. Temperature distribution inside the droplet is approximated by the second order polynomial (for example, see Dombrovsky and Sazhin [2003], Subramanian et al. [2001]): ...
... This clearly demonstrates that parabolic approximation is invalid, since at l t the quasi-steady parabolic profile does not yet exists. To relax this controversy, it has been suggested by Dombrovsky and Sazhin [2003] to artificially smooth the transient dependence s Tt at small times by applying the following ad hoc relation: ...
Two simple and yet sufficiently accurate approaches to predict surface temperature of a vaporizing droplet (higher order polynomial approximation and the heat balance integral method) are proposed. Being computationally inexpensive these approaches are tested as candidates for use in high-resolution LES spray modelling. Robust and efficient numerical algorithm for solving inherently stiff equations of droplet heating and evaporation has been developed. Robustness and computational efficiency of the proposed algorithm is achieved by use of unconditionally stable strongly implicit integration scheme and appropriate adaptation of the time step. The above methodology has been implemented in CFD spray model included in the in-house Fire3D code. The spray model has been applied to replicate three essentially different experimental scenarios in which turbulent sprays of water, acetone, and diesel fuel were investigated. Reasonable agreement has been demonstrated for predicted and measured droplet sizes and velocities as well as for the spray tip penetration dynamics. New numerical algorithms used to calculate surface temperatures of evaporating droplets with non-uniform internal temperature did not incur observable increase of CPU time in turbulent spray simulations.
... This clearly demonstrates that parabolic approximation is invalid at t ( s l , since the quasi-steady parabolic profile does not yet exists. To relax this controversy, it has been suggested in [10] to artificially smooth the transient dependence T s (t) at small times by applying the following ad hoc relation: ...
... Two-temperature approach can be derived more rigorously if a particular functional form of internal temperature distribution is assumed. This is the idea of parabolic approximation when the radial temperature distribution is assumed to be second order polynomial as suggested by Dombrovsky and Sazhin [10]. The droplet core temperature is formally defined as volume-averaged temperature which is coupled (via heat balance ODE) to droplet surface temperature and the heat flux received by the liquid. ...
... In this method, temperature distribution inside the droplet is approximated by the second order polynomial (for example, see [10,11]): ...
... where S h,d,i accounts for the energy transfer due to droplet breakup, convective heat transfer, and evaporation. Equation (23) does not include the energy diffusion term because a parabolic temperature profile model [68] considers the finite thermal conductivity within the droplet classes. Following the guidelines described in [68], the droplet surface temperature is calculated as: ...
... Equation (23) does not include the energy diffusion term because a parabolic temperature profile model [68] considers the finite thermal conductivity within the droplet classes. Following the guidelines described in [68], the droplet surface temperature is calculated as: ...
Advancements in internal combustion technology, such as efficiency improvements and the usage of new complex fuels, are often coupled with developments of suitable numerical tools for predicting the complex dynamic behavior of sprays. Therefore, this work presents a Eulerian multi-fluid model specialized for the dynamic behavior of dense evaporating liquid fuel sprays. The introduced model was implemented within the open-source OpenFOAM library, which is constantly gaining popularity in both industrial and academic settings. Therefore, it represents an ideal framework for such development. The presented model employs the classes method and advanced interfacial momentum transfer models. The droplet breakup is considered using the enhanced WAVE breakup model, where the mass taken from the parent droplets is distributed among child classes using a triangular distribution. Furthermore, the complex thermal behavior within the moving droplets is considered using a parabolic temperature profile and an effective thermal conductivity approach. This work includes an uncertainty estimation analysis (for both spatial and temporal resolutions) for the developed solver. Furthermore, the solver was validated against two ECN Spray A conditions (evaporating and non-evaporating). Overall, the presented results show the capability of the implemented model to successfully predict the complex dynamic behavior of dense liquid sprays for the selected operating conditions.
... En ce qui concerne la diffusion de la chaleur à l'intérieur de la goutte, l'une des possibilités, afin d'économiser du temps de calcul, notamment par comparaison à la résolution complète de l'équation de la chaleur dans la goutte, consiste à supposer la forme du profil de température à l'intérieur de la goutte, tout en conservant des hypothèses d'axi-symétrie. Dombrovsky et Sazhin [21] ont utilisés une approximation de type parabolique pour la distribution de température sur un rayon de la goutte : ...
... L'intégration des équations précédentes conduit à des profils paraboliques pour les fractions instantanées de la masse et de la température (Dombrovsky et Sazhin [21] et Dombrovsky et Sazhin [23]). ...
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... and the parabolic temperature profile in the particle just after solidification (at ) [63][64][65]: ...
... The so-derived approximate relations were recommended for implementation into FCI simulation codes. It is well-known that the parabolic model gives good results for quasi-steady heating or cooling of the particle with heat absorption or generation inside the particle without phase changes [63][64][65]. However, the parabolic model is not applicable to a solidifying melt droplet. ...
The chapter is concerned with radiation heat transfer modeling in multiphase disperse systems, which are formed in high-temperature melt-coolant interactions. This problem is important for complex interaction of the core melt with water in the case of a hypothetical severe accident in light-water nuclear reactors. A considerable part of thermal radiation emitted by the melt droplets lies in the range of water semi-transparency. As a result, the radiation is not completely absorbed in water and one needs to account for radiation heat transfer between the particles which have different temperatures. The scattering of radiation by steam bubbles and melt droplets separated from ambient water by a thin steam layer is also important. The problem is further complicated by semi-transparency of small oxide droplets and temperature differences between the center and surface of the melt droplets during their solidification. Nevertheless, the specific radiative properties of the multiphase flow components allow for a simplified approach, which is implemented in a problem-oriented CFD code. A more sophisticated approach for visible radiation of the multiphase media is also presented. The latter is expected to be important for optical diagnostics of the flow in small-scale experiments including those using various stimulant melts.
... A comprehensive review is provided in [61]. The variety of these models (e.g., parabolic model [62], power law model [63], polynomial model [64] and heat balance integral method [65]) stems essentially from the nature of the functional form that is used for the temperature distribution. In the parabolic approximation, a quasi-steady profile is assumed to be established immediately and the transient stage is ignored [61]. ...
... In the application of RR to the measurement of evaporating and burning droplets, a parabolic distribution of refractive indices within the droplet was usually assumed [14,15], as this type of distribution is close to that calculated using the effective diffusion model and thus satisfies the equation of thermal balance at the droplet surface. Vetrano et al. [16] presented an inversion algorithm for the determination of refractive index at the droplet surface and core. ...
Rainbow refractometry was used to measure the temperature and size of transparent spherical particles. In practice, however, there are limitations to the application of heating and cooling droplets, as the temperature measured is neither the average nor the surface or core temperature of the droplet. Reported here is an exploitation of this technique for droplet surface temperature determination. Droplet surface tension was measured by detecting the evolution of interference fringes of oscillating droplets. The dependence of surface tension on temperature facilitated the study of surface temperature of an evaporating droplet with time. Moving ethanol, n -heptane, and n -decane droplets were investigated under heating and cooling conditions. The capabilities and limitations of rainbow refractometry were verified by comparing the droplet temperature values measured directly by rainbow refractometry with the surface temperature.
... 3), or the one described in [36] where a parabolic approximation was offered for the temperature profile inside a symmetrically heated droplet (see Sect. 2.1.1.5). ...
The validity of the assumption that both liquid and gas phases can be treated as a continuum, used in the previous chapters, is questionable when the interface between liquid and gas is modelled, even when the gas pressure is well above atmospheric. The chapter starts with a review of early kinetic models of droplet evaporation. Then more rigorous models, using numerical solutions to the Boltzmann equations for vapour and air, are described. Two regions above the surface of an evaporating droplet are considered: the kinetic and hydrodynamic regions. Vapour and air dynamics in the first region are described by the Boltzmann equations, while the conventional hydrodynamic approach is used in the second region. Collisions between molecules are assumed to be inelastic in the general case. The evaporation coefficient is estimated using molecular dynamics analysis of n-dodecane molecules, based on the united atoms model. The applicability of quantum-chemical models to finding this coefficient is investigated.
... due to a capillary flow [43], [44] or convection [45], there is a radial distribution of SDS in the droplet and the concentration of SDS at the surface can be significantly higher than its mean concentration. It seems reasonable to assume [46], [16]] that the radial profile of the distribution is constant, while its local value changes along with the average SDS concentration in the droplet: . ...
We study the evolution of the mass of evaporating single microdroplets of sodium do-decyl sulphate (SDS) / diethylene glycol (DEG) mixture. First, we recognise and deconvolute the influence of residual water evaporation [Kolwas et al. Soft Matter 2019;15:1825], which accelerates the composite droplet evaporation, a simple exponential decay of the evaporating droplet surface change rate. This enables us to study the influence of SDS concentration on the composite droplet evaporation. Next, we establish a simple relationship between the average SDS concentration and the droplet evaporation rate to enable the study of the evolution of SDS concentration at the droplet surface. The oscillatory nature of surface SDS concentration indicates cyclic changes in the surface monolayer associated with the cyclic creation of vesicles (micelles) at the surface. The model we developed, allows the determination of SDS critical micelles concentration (CMC) in DEG as 60+/-2 mM.
... Regarding the diffusion of heat inside the drop, one of the possibilities, in order to save calculation time, in particular by comparison to the complete resolution of the equation of heat in the drop, consists in assuming the shape of the profile of temperature inside the drop, while retaining axi-symmetry assumptions. Dombrovsky and Sazhin used a parabolic-type approximation for the temperature distribution over a radius of the drop [12]. The problem of droplet evaporation can appear relatively complex when all of the physical phenomena involved in the exchanges between the droplet and the gas phase are taken into account in the modeling. ...
In this work, it proposed an appropriate form of the droplets velocity formula in the integral form of the spray moment conservation equations, and coupling the effects of relative velocity term on the droplet heat transfer model in pressure-swirl atomized sprays. The spray tip penetration calculated in this approach agrees well with experimental data. The revised treatment of therm-physical properties of the liquid and gas phases also leads to quite stable calculations for a wide range of ambient temperatures and density ratios. The spray moments theory makes it possible to describe polydisperse sprays using an Eulerian approach and therefore appears to be a method indicated for two-phase evaporation applications. Its relevance for simulation at the scale of industrial applications is assessed in this work, by its implementation in two-dimensional configuration more representative of this type of pressure-swirl simulations. This evaluation couples a feasibility study in terms of calculation cost with an analysis of the precision obtained, by comparisons with the experimental data of reference methods for the description of sprays.
... To capture more details regarding droplets' internal thermal behaviour, e.g. finite thermal conductivity and internal recirculation within the droplets, the developed model uses a parabolic temperature profile model [35] together with the effective thermal conductivity model [36]. We are currently adding a species transfer equation which is the last prerequisite for adding single component evaporation capability. ...
... Later, Lick [26] and Nield [27] studied the effect of piecewise profile for temperature gradient on the onset of convection. The parabolic temperature profile model for convective heating of droplets is considered in Dombrovsky and Sazhin [28]. In the past, situations have been encountered similar to blast furnace envelopes which experience extreme temperature gradients and a non-linearity of temperature profile is expected (see Ficker et al. [29]) . ...
In the present study, the effects of different types of basic temperature and concentration gradients on a layer of reactive fluid under variable gravity field are analyzed using linear and non-linear analysis. Energy method is applied to obtain the non-linear energy threshold below which the solution is globally stable. It is found that the linear and non-linear analysis are not in agreement for the considered models of temperature and concentration gradients. The obtained results of non-linear analysis for different values of reaction terms and variable gravity coefficients in each given model of temperature and concentration gradients are compared with the linear instability results. The Chebyshev pseudospectral method is used to obtain the numerical and graphical results of subsequent analysis.
... We have now all the data to estimate the maximum temperature difference, T gp , between the surface of the glass plate and the plane of symmetry of this plate. In the quasi-steady regime of the plate heating, the temperature profile is very close to the parabolic one, and it is sufficient to use a parabolic approximation for the temperature profile in the plate as it was done in [34] . The parabolic profile is a self-similar solution of the conduction equation. ...
It is confirmed experimentally that both the heat conduction along the substrate and its heat capacity affect significantly the self-organized structures on the surface of an evaporated polymer film. The analysis presented is based on a combination of the laboratory experiments and heat transfer modeling. It is shown that only a part of the holes on the film surface can be classified as the breath figures, whereas the others are formed due to the solvent evaporation in the film volume, under the solid surface crust of the film. A combined experimental and computational analysis enables the authors to obtain a convective instability of the boundary layer flow triggered by the fast evaporation of the solvent. Most likely, this effect is responsible for the large-scale surface pattern which looks as vertical strips of Bénard-like cells. The orifices grouped at the boundaries of these convective cells are definitely produced by the volume evaporation of solvent under the surface crust. The suppression of surface patterns with the use of metal-containing substrates is accompanied by more intense producing the porous structure inside the polymer film.
... Yuen [3] analyzed the change of the resistance of the droplet among the evaporation process with Reynolds number from 10 to 100. Dombrovsky et al. [4] studied effects of the finite thermal conductivity, assuming that the internal temperature of the droplet was distributed in parabolic, and they proposed an evaporation model which was applicable to the convective heat transfer of liquid droplets. Based on the above model, Abramzon [5] further included the radiation heat transfer of liquid droplets. ...
In the cryogenic wind tunnel, cooling the circulating gas to cryogenic temperature by spraying liquid nitrogen (LN2) is an efficient way to increase the Reynolds number. The evaporation and motion of LN2 droplets in the high-speed gas flow is the critical process that determines the cooling rate, cooling capacity and the safe operation of the down-stream compressor. In this study, a numerical model of droplet motion and evaporation in high-speed gas flow is developed and verified against experimental data. The droplet evaporation rate, diameter and velocity are obtained during the evaporation process under different gas temperatures and flow velocities. The results show that the gas temperature has dominant influence on the droplet evaporation rate. High flow speed can increase droplet evaporation effectively at the beginning process. Evaporation of droplets with different diameters follows a similar trend. The absolute evaporation rate increases with the increase of droplet diameter while the relative evaporation amount is highest for the smallest droplet due to its high area-volume ratio. This numerical study provides insight for understanding the evaporation of LN2 droplets in high-speed gas flow and useful guidelines for the design of LN2 spray cooling.
... Alternatively, heat and mass transfer is closely related to both droplet surface temperature and average temperature. Dombrovsky and Sazhin [17] developed a parabolic temperature profile model, in which the internal temperature was approximated to a parabolic form. This model was less accurate than either the conduction limit or effective thermal conductivity model, but offered comparable computational efficiency to the lumped model. ...
Droplet heating and evaporation is of fundamental importance in various engineering applications. Based on the assumption that droplets’ internal temperature follows a third-order polynomial distribution, a new liquid phase model is proposed, taking into account both temperature gradient and internal circulation inside the droplet in terms of Hill’s spherical vortex. The new liquid model and other commonly used liquid models are compared with the experimental data outlined in the literature. The internal temperature predicted by the new model matches the experimental data of an n-Decane droplet heating best, whereby the enhancement of heat transport by internal circulation is correctly reflected. In comparison, other models always predict the monotonic variation of internal temperature during the initial heating period, which deviates widely from the experimental data. The new model displays the best performance with regard to predicting droplet evaporation in the later vaporizing period, whereas the parabolic temperature profile model and conduction limit model significantly underestimate and overestimate the droplet evaporation rate, respectively. The CPU requirement for the new model is far less than that for the conduction limit model and the effective thermal conductivity model with an order of magnitude, which shows its potential for implementation into CFD codes.
... Following this approach, these strict assumptions were relaxed by taking into consideration the relative velocity between the air and the droplet, the transient droplet heating, the Stefan flow effects, the evaporation of multicomponent droplets and other secondary phenomena. The transient droplet heating was accounted with the Infinite Conductivity Model (ICM) [11] and the spatial temperature distribution inside the droplet with the Finite Conductivity Model (FCM) [12], the Effective Conductivity Model (ECM) [13] and the parabolic temperature profile model [14]. Regarding the Stefan flow effects which arise from the radial vapor motion, the models of Abramzon & Sirignano [13] and Yao et al. [15] are widely used, while there is a large variety of available heat/mass transfer correlations to account for the relative droplet-gas motion [7]. ...
This paper presents CFD predictions for the evaporation of nearly spherical suspended droplets for ambient temperatures in the range 0.56 up to 1.62 of the critical fuel temperature, under atmospheric pressures. The model solves the Navier-Stokes equations along with the energy conservation equation and the species transport equations; the Volume of Fluid (VOF) methodology has been utilized to capture the liquid-gas interface using an adaptive local grid refinement technique aiming to minimize the computational cost and achieve high resolution at the liquid-gas interface region. A local evaporation rate model independent of the interface shape is further utilized by using the local vapor concentration gradient on the droplet-gas interface and assuming saturation thermodynamic conditions. The model results are compared against experimental data for suspended droplet evaporation at ambient air cross flow including single- and multi-component droplets as well as experiments for non-convective conditions. It is proved that the detailed evaporation process under atmospheric pressure conditions can be accurately predicted for the wide range of ambient temperature conditions investigated.
... Greek symbols a thermal diffusivity (m 2 /s) Regarding the evaporation modeling, several simplified 0-D and 1-D models have been proposed to predict the evaporation of isolated spherical droplets. Starting from the classical ''D 2 -law" of Godsave [24] and Spalding [25], various approaches have been proposed to include the transient droplet heating by prescribing the internal temperature distribution as in [26][27][28][29][30][31] and the Stefan flow effects as in Abramzon and Sirignano [28] and Yao et al. [32]. The performance of these models was assessed in comparative studies such as those of [33][34][35][36] and defined their range of applicability. ...
The Navier–Stokes equations, energy and vapor transport equations coupled with the VOF methodology and a vaporization rate model are numerically solved to predict aerodynamic droplet breakup in a high temperature gas environment. The numerical model accounts for variable properties and uses an adaptive local grid refinement technique on the gas–liquid interface to enhance the accuracy of the computations. The parameters examined include Weber (We) numbers in the range 15–90 and gas phase temperatures in the range 400–1000 K for a volatile n-heptane droplet. Initially isothermal flow conditions are examined in order to assess the effect of Weber (We) and Reynolds (Re) number. The latter was altered by varying the gas phase properties in the aforementioned temperature range. It is verified that the We number is the controlling parameter, while the Re number affects the droplet breakup at low We number conditions. The inclusion of droplet heating and evaporation mechanisms has revealed that heating effects have generally a small impact on the phenomenon due to its short duration except for low We number cases. Droplet deformation enhances heat transfer and droplet evaporation. An improved 0-D model is proposed, able to predict the droplet heating and vaporization of highly deformed droplets.
... Great efforts have been made to improve the computational efficiency for modeling the droplet heating and vaporization processes [9,10]. Dombrovsky and Sazhin [11] used the parabolic temperature profile model with the presumed parabolic temperature distribution within the droplet to simulate the heat transfer process in the liquid phase for the single-component droplets. Ra et al. [10] developed a simplified surface temperature sub-model, in which the difference between the droplet surface and core temperatures is considered, thus the heat transfer within the droplet can be simulated. ...
A new quasi-dimensional multi-component vaporization model considering the finite thermal conductivity and mass diffusivity within the droplet was constructed. First, the heat flux of conduction, enthalpy diffusion, and radiation absorption in the gas phase were calculated based on the Fourier’s law, a multi-diffusion sub-model, and a simplified analytical solution, respectively. The phase equilibrium at the gas–liquid interface was calculated by the ideal and real gas approaches according to the ambient pressure. For the liquid phase, the assumption of the quadratic polynomial distributions of the temperature and component concentration within the droplet was proposed in the quasi-dimensional model. Then, the proposed vaporization model was extensively validated by the experimental measurements, and good agreements were observed. Based on the computational results, the vaporization and movement behaviors of fuel droplets under forced convection conditions were further understood. Finally, by comparing with the zero-dimensional vaporization model with uniform temperature and component concentration distributions within the droplet and the one-dimensional vaporization model with finite thermal conductivity and mass diffusivity in the radical direction of the droplet, it is found that the quasi-dimensional model agrees better with the one-dimensional model than the zero-dimensional model, especially for the conditions with high ambient temperature and velocity. Sensitivity analysis indicates that the temperature gradient within the droplet plays a significantly important role in the droplet vaporization process.
... Here we use a finite conductivity model. The temperature profile between the droplet surface and its center is assumed to be a parabola [20]: ...
MILD Combustion, also known as flameless combustion, is attracting wide scientific interest due to its potential of high efficiency and low NOx emission. The Delft Spray-in-Hot-Coflow (DSHC) burner was designed to study the fundamental aspects of flameless oxidation of light oils. The present paper reports a numerical study of an ethanol flame of the DSHC database with the transported PDF method. A Lagrangian-Lagrangian description is used for the two-phase turbulent flow system in the dilute spray region. The continuous phase is modeled with a joint velocity-scalar PDF. And the dispersed phase is described by a joint PDF of droplet parameters. To cope with the high-dimensionality, the joint PDFs are solved by a Monte Carlo particle method. The evolution of the gas phase composition is described by a Flamelet Generated Manifold (FGM) and the IEM micro-mixing model. The droplet heating and evaporation processes are modeled with a finite conductivity model taking into account the effects of Stefan flow. Validation of this modeling approach is carried out by comparison with experimental measurements. Results show that the current modeling approach is capable of accurately predicting main properties, including dispersed phase and gas phase mean velocity, and droplet Sauter Mean Diameter. Gas phase temperature is also in reasonable agreement with the measured data; however it is under predicted in the region where the enthalpy loss due to droplet evaporation has a non-negligible effect. Improvement of the accuracy of temperature prediction can be made by using a non-adiabatic FGM table including an enthalpy variable.
... где d -характерный диаметр сферической частицы среды 1. Соотношение (3) получено с использованием параболической аппроксимации профиля температуры в частице [13,14]: ...
... and the parabolic temperature profile in the particle just after solidification [41]: ...
The recently developed model for thermal radiation in multiphase flows typical of melt-coolant interactions is generalized to account for transient temperature profile in large semi-transparent particles of solidifying melt. A modification of the Large-Cell Radiation Model (LCRM) is based on approximate solution for coupled radiation and conduction in optically thick spherical particles of a refractive material. The simplicity of the suggested approximation enables one to implement the modified model in a multiphase CFD code. The LCRM extension makes possible the use of this approach not only for the core melt in nuclear fuel-coolant interactions (FCI’s) but also for other melt substances which are widely used in the laboratory experiments. The numerical data demonstrate an effect of absorption coefficient of the particle substance on the rate of particle cooling and solidification.
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... -Modèles faisant appel à la résolution des équations de Navier-Stokes 2D ou 3D à l'intérieur de la goutte. Le modèle "parabolic temperature profile" proposé par Dombrovsky et Sazhin [59,177], qui correspond à un modèle à conductivité thermique finie, permet notamment d'améliorer significativement les résultats par rapport au modèle de Spalding, tout en limitant le surcoût associé à la prise en compte de la non-uniformité de la goutte. D'autres modèles sont disponibles dans la littérature mais ne sont pas détaillés ici [177,188]. ...
The combustion of hydrocarbons still represents the major part of the worldwide production of energy, especially for aerospace. Most industrials burners are fed with liquid fuel that is directly injected in the combustion chamber, generating a strong interaction between the spray, the turbulent flow and the combustion. This interaction has been widely studied, but is not yet fully understood. In particular, modeling individual droplet combustion, in the framework of Large Eddy Simulation (LES) of complex geometries, is a difficult issue. This work aims at improving models for spray combustion, in the context of two-phase reactive LES of complex configurations using an Euler-Lagrange approach. First, a droplet combustion model accounting for the various regimes and called MustARD for « Multi-State Algorithm for Reacting Droplets » is proposed and validated on several academic configurations of growing complexity. Second, MustARD is evaluated in the LES of a lab-scale burner and compared to classical models neglecting individual droplet combustion. Results show in particular the importance of the new model and its impact on the flame structure. Moreover, the comparison with experiments shows that MustARD contributes to improve the numerical prediction of LES of two-phase reacting flows.
... where d is the characteristic diameter of a conventional spherical particle of medium 1. Equation (3) was obtained with the use of the following parabolic approximation of temperature profile in the particle [19,20]: ...
... Temperature distribution inside the droplet is approximated by the second order polynomial (for example, see [11, 12]): ...
The purpose of this work is to analyze the importance of considering internal temperature gradient in modeling droplet evaporation, and to demonstrate performance of simplified methods in which the temperature gradient is approximately taken into account. Based on three characteristic time scales, two dimensionless criteria have been identified which determine magnitude of the internal temperature gradient and its effect on the evaporation dynamics. Numerical values of these criteria in a wide range of ambient temperatures show that the effect of the internal temperature gradient is more pronounced in more volatile liquid at higher ambient temperatures. Although droplet life time predictions are not sensitive to the internal temperature gradient, its effect might be strong at the initial stages of droplet evaporation, and this substantiates the need in robust and computationally inexpensive methods to take it into account. Two simple and yet accurate approaches (quasi-steady higher order polynomial approximation and the integral balance method) have been favourably tested and recommended for use in CFD spray modeling.
... Assuming a parabolic temperature profile inside the particle (Dombrovsky and Sazhin, 2003), the energy equation for the particle average temperature, T p , can be written as: ...
In this study, via an Eulerian–Lagrangian framework, the performance of two recent dispersion models, i.e. a first-order autoregressive process and the PDF model, is compared. The appropriate relations for the turbulence scales and the drift correction term are suggested and the tuned values for the constants of the models are proposed in a systematic approach by starting with the simplest case, i.e. particle-laden stationary isotropic turbulence and adding more complexities in the subsequent cases, including the homogeneous anisotropic shear flow, decaying grid turbulence, and inhomogeneous gas–solid spray. Also, the isotropic relation for the effect of inertia in the Lagrangian turbulence time scale seen by particles is extended to the anisotropic case while it remains consistent in the isotropic limit. Finally, the performance of the tuned models is evaluated for the simulation of an evaporating spray. It is observed that, the tuned constants for the evaporating spray are close to the ones obtained for the homogeneous shear flow.
Localized heating of hydrophobic droplet is considered and the effects of restricted area heating and droplet size on heat transfer are examined. Samples composing of vertical steel pins located in Perspex holders are designed and manufactured. Sample surfaces are hydrophobized via depositing treated silica nanoparticles by dip coating method. Hydrophobicity of pin and Perspex surfaces are evaluated simultaneously securing uniform wetting state over sample surface. Droplet is heated via pin while creating a localized heating effect on droplet fluid. The experiments are conducted obtaining flow structures in droplet liquid. Simulations are also performed predicting thermal state in droplet liquid during heating while adopting conditions of the experiments. It is found that hydrophobizing of samples results in uniform contact angle (150° ± 2°) over the entire surface having hysteresis of 6° ± 3°. Enlarging pin diameter and droplet size alter flow structures inside droplet; hence, center of circulating structures changes, which modify the ratio of convection over conduction currents at droplet liquid interface. Increasing pin diameter enhances the Nusselt and the Bond numbers, which becomes more apparent as droplet volume increases.
Droplet evaporation plays a decisive role in determining the evolutions of droplet temperature, diameter, and velocity in sprays with low saturation temperature and high volatile mediums. This study conducts a comparative analysis of liquid phase models for single droplet heating and evaporation of high volatile R134a. Three traditional liquid phase models based on the assumption of infinitely large liquid thermal conductivity (lumped), limited thermal conductivity, and effective thermal conductivity and a recently proposed third-order polynomial temperature profile (third-order) liquid phase model based on the presentation of the temperature profile inside the droplet in cubic polynomial form without solving the heat conduction equation are introduced to predict the temporal evolutions of droplet temperature, diameter, and velocity, and the spatial evolution of temperature within the droplet. The coupling of heat and mass transfer between the droplet and its surrounding gas is incorporated into the liquid phase models to investigate its effect on droplet evaporation. The droplet temperature first undergoes a rapid decrease at the transient stage of evaporation and then decreases slowly to the minimum temperature (Tmin). The droplet temperature in the transient stage depends highly on the choice of liquid phase models. Tmin is mostly unaffected by the liquid phase models. The dependence of the droplet diameter at the transient stage of evaporation on the choice of the liquid phase model is stronger compared with the following steady stage. The droplet velocity is rarely influenced in the whole evaporation. The liquid phase models incorporating the coupling weaken the evaporation intensity, resulting in a high droplet temperature and large droplet diameter. The third-order liquid phase model incorporating the coupling model predicts Tmin and agrees with the experimental data. This model can reflect reasonably the heat transfer enhancement by the recirculation inside the droplet for moving droplet.
We study the evolution of the mass of evaporating single microdroplets of sodium dodecyl sulphate (SDS) / diethylene glycol (DEG) mixture. First, we recognise and deconvolute the influence of residual water evaporation [Kolwas et al. Soft Matter 2019;15:1825], which accelerates the composite droplet evaporation, a simple exponential decay of the evaporating droplet surface change rate. This enables us to study the influence of SDS concentration on the composite droplet evaporation. Next, we establish a simple relationship between the average SDS concentration and the droplet evaporation rate to enable the study of the evolution of SDS concentration at the droplet surface. The oscillatory nature of surface SDS concentration indicates cyclic changes in the surface monolayer associated with the cyclic creation of vesicles (micelles) at the surface. The model we developed, allows determination of SDS critical micelles concentration (CMC) in DEG as 60±2 mM.
A numerical model for evaporation along with internal radiation absorption of a non-spherical droplet is presented. In this model, the VOF method is utilized to trace the interface and the radiative transfer is taken into account by using the Monte Carlo method combined with a contour based interface reconstruction algorithm. Meanwhile, the dynamic mesh technique is applied to increase the computational efficiency. The validated model is used to investigate the radiation absorption inside prolate and oblate water droplets under infrared laser irradiation, and a parametric study for the effects of droplet’s shape and size is presented. The results reveal that the shape of droplet dominates the spectral absorptance when the droplet’s optical thickness is small, but for optically thick droplets, the radiation absorption turns to be dominated by the spectral reflectivity of the irradiated surface. For a deformable droplet under asymmetrically radiative heating, the incident radiation can significantly enhance the heat transfer and evaporation processes. In addition, the assumption of uniform radiation absorption is found to be valid to optical thin droplets, whereas this assumption will underestimate the average interface temperature and evaporation rate of droplets with large optical thickness.
A novel two-zone model is implemented, within an in-house code, for the heating of an evaporating droplet. While it is not yet possible to fully assess and validate the novel model due to lack of available detailed experimental data, the concept is described. Differences compared to the ‘common’ isothermal model are explained. Firstly, a quantitative assessment is performed, using experimental datasets where a suspended water droplet is exposed to a hot air flow. Secondly, the behavior of the novel model is examined over a wider range of conditions in terms of air velocity, ambient temperature and initial droplet diameter.
The novel model predicts droplet lifetimes and saturation temperatures that are similar to the isothermal model. Nevertheless, the results also show that, during the early stages, the cooling of the surrounding gas due to the evaporation of the droplet is more pronounced with the two-zone model. During a second stage, the evaporation term is slightly lower than with the isothermal model. The available data do not allow to fully assess which result is closer to reality, but the important observation is that there exist differences. These differences increase with higher ambient temperature, relative velocity or initial diameter.
Spray reactive flow finds application in various technical devices, and due to the complex nature, their optimization is very challenging, requiring proper modeling of turbulence/chemistry interactions as well as of the contribution from spray evaporation. This work presents a study of sub-grid scale combustion models, where relevant assumptions on multiphase coupling and their effects are analyzed in detail. For this purpose, two different flamelet approaches, i.e. progress variable spray flamelet and multi-regime gas flamelet are examined in an implementation coupled with the dynamic thickened flame model, along with which the impact of inlet inhomogeneities condition and droplet evaporation taking into account internal temperature gradient is also investigated. The numerical evaluation is carried out in large eddy simulations of a benchmark ethanol spray flame with partial pre-vaporization, where an Eulerian-Lagrangian numerical framework is adopted. The analysis demonstrated that the flame dynamics under consideration is governed by a close coupling between spray evaporation, turbulent dispersion and unsteady flame propagation at upstream shear layers. Results show that the spray flamelets built from counterflow partially-premixed spray flames achieved a better agreement with experiments, capturing the flame structure in terms of gas-phase temperature, OH mass fraction as well as spray statistics.
Internal fluidity of a sessile droplet on a hydrophobic surface and dynamics of fine size dust particles in the droplet interior is examined for various droplet contact angles. The geometric features of the droplet incorporated in the simulations resemble the actual droplet geometry of the experiments. Simulation conditions are set in line with the experimental conditions. The dust particles are analyzed and the surface tension of the fluid, which composes of the dust particles and water, is measured and incorporated in the analysis. Particle tracking method is adopted experimentally to validate the numerical predictions of the flow field. It is found that heat transfer from the hydrophobic surface to the droplet gives rise to the formation of two counter rotating cells inside the droplet. The Nusselt and the Bond numbers increase with increasing droplet contact angle. The number of dust particles crossing over the horizontal rake, which corresponds to the top surface of the dust particles settled in the droplet bottom, towards the droplet interior increases as the particle density reduces, which is more pronounced in the early period. Experimental findings of flow velocity well agree with its counterparts obtained from the simulations.
A multicomponent droplet vaporization model which combines the computational efficiency of continuous thermodynamic approaches with the detailed species information provided by discrete component models has been developed. The Direct Quadrature Method of Moments (DQMoM) is used to efficiently solve for the evolution of the nodes and weights of the equivalent liquid-phase mole fraction distribution without assuming any functional form. The novelty of the approach is an inexpensive delumping procedure that is used to reconstruct the time-dependent mole fractions and fluxes for all discrete species. When applied to a vaporizing kerosene droplet, agreement between the full discrete component model, which solves ODEs for every individual species, and DQMoM with delumping, which solves only a few ODEs, is excellent. This computationally inexpensive model is well-suited for implementation in CFD codes with detailed kinetic mechanisms, as it enables accurate calculation of species source terms from the droplets without incurring an unrealistic computational cost.
The validity of the assumption used in the previous chapters that both liquid and gas phases can be treated as a continuum is no longer obvious when the interface between liquid droplets and the ambient gas is modelled, even when the gas pressure is well above one atmosphere. The chapter begins with a review of early kinetic models of droplet evaporation. Then more rigorous models, based on numerical solutions to Boltzmann equations for vapour and air, are discussed. Two regions of gas above the surface of an evaporating droplet are considered: the kinetic and hydrodynamic regions. Vapour and air dynamics in the first region are described by the Boltzmann equations, while the conventional hydrodynamic analysis is applied in the second region. Collisions between molecules are assumed to be inelastic in the general case. The evaporation coefficient is estimated based on molecular dynamics analysis of n-dodecane molecules, using the united atoms model (bonding between hydrogen and carbon molecules is assumed to be much stronger than bonding between carbon molecules).
The work aims at developing and setting up a model able to predict the diesel spray evolution to be integrated into a complete thermodynamic model of the engine. Previous papers have been devoted to realize a numerical model for the injection system, in which a lumped parameter + one-dimensional approach is employed. Such a model has been now enhanced by introducing a quasi-dimensional model for fuel break up, diffusion and penetration processes. A self-developed heating sub-model is included in the model, which enables the evaluation of the influence of the fuel properties on the evaporation process. As a result, the injection system simulation model gives indications on the spray formation process and it is used into a lumped parameter model of the combustion process. Results concerning the influence of fuel properties on the evaporation process are presented and discussed, pointing out its effect on engine performance.
A previously developed injection system model has been enhanced including a quasi-dimensional, multi-zone, direct injection (DI) diesel combustion model, with the aim of taking into account the actual injection process, the spray formation and the droplet heating-vaporization processes. Such a goal is obtained by means of the integration of different modeling approaches. In a commercial simulation environment, a lumped parameter mechanical-hydraulic scheme is used to model the injection process, in terms of fuel flow rate and injection pressure. The spray formation processes and the droplet vaporization phenomena are then implemented in a self developed computation code, accounting for finite thermal conductibility of the liquid phase fuel. The coupling among the models allows for a detailed representation of the involved phenomena at each simulation step (e.g. fuel time pressure history, fuel properties, atomization, evaporation ambient condition); at the same time, it is possible to evaluate the operation of the injection system on the basis of atomization, vaporization and combustion behavior. The results of the numerical prediction are compared to experimental data referred to a DI diesel engine.
The performance of the ensemble diesel engine-injection system has been investigated by modeling; different approaches have been combined and integrated in a multi step procedure. In a preliminary phase, the nozzle flow features have been investigated by means of 3D CFD simulation (FIRE), allowing for the evaluation of nozzle hole discharge coefficients. Such values have been used in the subsequent phase, based on a 0/1D approach, in which the complete model of the fuel injection system has been realized and the injected fuel amounts have been evaluated. On the basis of these indications, a further simulation phase has been included and a packet model for the modeling of fuel jet formation, evaporation and combustion has been built and used. In-cylinder pressure trends have been obtained and the effects of different nozzle hole geometries have been investigated.
Simple approximate formulae describing temporal evolution of diesel fuel droplet radii and temperatures predicted by the kinetic model are suggested. These formulae are valid in the range of gas temperatures relevant to diesel engine-like conditions and fixed values of initial droplet radii, or in the range of initial droplet radii relevant to diesel engine-like conditions and fixed values of gas temperature. During the time period before the hydrodynamic model predicts complete droplet evaporation, these approximations are based on the calculation of the correction to the prediction of the hydrodynamic model. At longer times, the approximations at the earlier times are extrapolated up until the total evaporation of the droplet, using quadratic fittings. The new approximations are shown to be reasonably accurate for predicting the temporal evolution of droplet radii and droplet evaporation times. The predictions of droplet temperature turned out to be less accurate than those of droplet radii, but this accuracy is believed to be sufficient for many practical applications.
In this article a new combination of evaporation sub-models is established for the evaporating/combusting many-droplet fuel spray by investigating different correlations for the heat and mass transfer rates. Then the model is used for the simulation of turbulent evaporating sprays via a two-way coupled Eulerian–Lagrangian framework. A turbulence closure is achieved based on the Reynolds stress transport model for the carrier gas phase with a calibrated dispersion model for the dispersed droplets. After the validation of the model by a recent experimental database, the effects of physical parameters such as the pilot air temperature, coflow air velocity, jet turbulence intensity and Sauter mean diameter of the injected spray on the evaporation delay of realistic evaporating polydispersed turbulent fuel sprays are carefully investigated. It is observed that the Sauter mean diameter has major effect but the turbulence intensity has minor effect on the evaporation delay.
More than 450 thermal contact conductance data points obtained from isotropic conforming rough surfaces for five different materials; nickel, stainless steel, two zirconium alloys, and aluminum have been compared with the existing elastic and plastic models. For the first time data have been reduced to a dimensionless form assuming both elastic as well as plastic deformation. Normally, data were compared with either the elastic model or the plastic model assuming a type of deformation a priori. The relative merits of different models and the surface factors influencing the mode of deformation are still not clear. Hence, the aim of the present work was to compare most of the models available in the literature with themselves as well as with isotropic data. Comparison showed that generally smoother surfaces deform elastically, and rougher ones plastically. However, there are some data sets that compare well with both the elastic as well as the plastic models.
The system of equations describing the effects of heating, evaporation, and combustion of fuel droplets in a monodisperse spray is simplified assuming that the Nusselt and Sherwood numbers are equal to 2. The radiative energy exchange between fuel droplets surface and gas is described by using the P-1 model with Marshak boundary conditions. The chemical term is presented in the Arrhenius form with the pre-exponential factor calculated from the enthalpy equation, using the Shell autoignition model. The resultant, singularly perturbed system of ordinary differential equations is analyzed, based on the geometrical version of the integral manifold method. The ignition process is subdivided into two stages: droplet evaporation and ignition of the gaseous mixture. Results predicted by the analytical solutions are compared with those predicted by the CFD package VECTIS. It is suggested that the analytical solution underpredicts the evaporation time. A considerably better agreement between the evaporation times predicted by VECTIS and the proposed theory is achieved when the gas temperature is assumed to be equal to the local temperature in the vicinity of droplets. The effects of thermal radiation are significant, especially at high temperatures and with large droplets, and cannot be ignored.
The study of droplets and sprays has developed rapidly over the past two decades because of their many important applications, from automobile engine combustion to drug aerosols. This book addresses the complex subject of the interactions of droplets and sprays. Along with a strong theoretical foundation, the book presents results in a way that will be useful for engineering practice, with summaries of key formulae and examples of various spray computations. Among topics covered are transient heating (or cooling) and vaporization (or condensation), multicomponent liquid droplet vaporization, near critical and supercritical ambient conditions, interaction of droplets with turbulent or vortical structures, distortion of the spherical shape and secondary atomization of the droplets, and computational issues. As an authoritative review of the science and technology of droplets and sprays, this book will be useful for graduate students, researchers, and practising engineers.
An analysis of regularly spaced disk sources on the surface of a semi-infinite body is given and related to steady-state contact conductance theory. It is shown that simple superposition utilizing the steady-state temperature distribution for a single typical disk source is not valid since a steady state does not exist for the temperature resulting for an infinite number of regularly spaced sources on the surface of a semi-infinite solid. A novel analysis is presented that treats the transient surface temperature in such a manner that a steady-state conductance is derived. The conductance results are compared with those obtained by M. M. Yovanovich who uses a complementary analysis. The method of analysis can be applied to other disk spacings and to random distribution of contacts. Also considered is the case of contact radius being a uniformly-distributed random variable which yielded the results of increased contact resistance compared to that using the average contact radius.
This book serves as both a graduate text and a reference for engineers and scientists exploring the theoretical and computational aspects of the fluid dynamics and transport of sprays and droplets. Attention is given to the behavior of individual droplets, including the effects of forced convection due to relative droplet–gas motion, Stefan convection due to the vaporization or condensation of the liquid, multicomponent liquids (and slurries), and internal circulation of the liquid. This second edition contains more information on droplet-droplet interactions, the use of the mass-flux potential, conserved scalar variables, spatial averaging and the formulation of the multi-continua equations, the confluence of spatial averaging for sprays and filtering for turbulence, direct numerical simulations and large-eddy simulations for turbulent sprays, and high-pressure vaporization processes. Two new chapters introduce liquid-film vaporization as an alternative to sprays for miniature applications and a review of liquid-stream distortion and break-up theory, which is relevant to spray formation.
Thermal contact conductance of nominally flat surfaces in contact was considered. The emphasis of the work is on effect of the mode of deformation on the value of conductance. Explicit expressions for thermal conductance were derived for cases of: (1) Pure plastic deformation (2) plastic deformation of the asperities and elastic deformation of the substrate and (3) pure elastic deformation. The last two important cases are considered for the first time here. Criteria which determine mode of deformation is also presented.
In recent work it has been shown that many types of surfaces used in engineering practice have a random structure. The paper takes, as a representation of the profile of such a surface, the waveform of a random signal; this is completely defined by two parameters, a height distribution and an auto-correlation function. It is shown how such a representation can be transformed into a model, appropriate for the study of surface contact, consisting of an array of asperities having a statistical distribution of both heights and curvatures. This theory is compared with the results of an analysis of surface profiles presented in digital form. The significance of these findings for the theory of surface contact and for the measurement and characterization of surface finish is discussed.
Numerical comparisons of the Greenwood-Williamson (GW) elastic microcontact model with two more general isotropic and anisotropic models suggest that the GW model gives good order-of-magnitude estimates of the number of contacts, real contact area fraction and nominal pressure that result at a given separation of a rough and a smooth flat plane.In effecting the comparisons, the three parameters of the GW model are related to the three spectral moments m0, m2 and m4 of an isotropic or equivalently isotropic surface.It is shown that the real contact area at a given mean plane separation depends only on the bandwidth parameter , while the elastically supported load depends on both α and m2.On the basis of the comparisons it is suggested that the GW model be adopted by metrologists to relate measured microgeometry to physically more understandable quantities such as contact density, load, real area and plastic contact density. Accordingly, this paper includes an expository presentation of the GW model complete with tables and a detailed numerical example of their use.
The elastic contact of an isotropically rough surface with a plane is treated by approximating the summits of a random process model by paraboloids with the same principal curvatures and applying the classical Hertzian solution for their deformation. The errors in this approximation are computed in terms of the separation and of a bandwidth parameter α. Load and real contact area are derived as functions of separation. For large separation the fractional area of real contact is found to be half the bearing area fraction and the separation may be eliminated to give direct porportionality between load and area. For all separations the load is approximately proportional to the contact area. The constant of proportionality depends only on the Hertzian elastic modulus and the profile absolute mean slope. Experimental measurements of the latter and of α for a variety of surfaces show little variation in either, and the measured values of α are within the range of applicability of the model. The theory is critically compared with existing theories.
Despite substantial theoretical studies of thermal contact conductance in the past, the application of statistical mechanics in this field has never been attempted. This paper addresses contact conductance from macroscopic and microscopic viewpoints in order to demonstrate the promise of the statistical mechanics approach. In the first part of the derivation, the Boltzmann statistical model is applied to determine the most probable distribution of asperity heights for a homogeneously, isotropically rough surface. The result found is equivalent to Gaussian distribution, which has only been assumed but not rigorously substantiated in the past. Subsequently, the Boltzmann statistical model is applied to predict the distribution of true contact spots when two such surfaces are pressed together, resulting in a relationship between the total thermal contact conductance and the relative interfacial pressure. The numerical results are compared to published empirical data, and a good order-of-magnitude agreement is found.
The characteristics of thermal contact conductance are increasingly important in a wide range of technologies. As a consequence, the number of experimental and theoretical investigations of contact conductance has increased. This paper reviews and categorizes recent developments in contact conductance heat transfer. Among the topics included are the theoretical/analytical/numerical studies of contact conductance for conforming surfaces and other surface geometries; the thermal conductance in such technological areas as advanced or modern materials, microelectronics, and biomedicine; and selected topics including thermal rectification, gas conductance, cylindrical contacts, periodic and sliding contacts, and conductance measurements. The paper concludes with recommendations for emerging and continuing areas of investigation.
The authors present some spray calculations in a combustion bomb and in a d.i. diesel engine for different test cases varying the formulation of the evaporation model in a modified version of the 3D-fluidynamic KIVA II code. Some numerical tests cases have been performed in a combustion bomb to analyze the influence of the modified evaporation model on droplet heating, evaporation rate and droplet lifetime. The numerical results have been compared with those obtained from the original Spalding model and with some experimental data obtained by an optical technique. In a second set of calculations, the new model has been tested in the combustion chamber of a real d.i. diesel engine. Results in terms of pressure, vapour mass, spray distribution and burned fuel are reported. A preliminary analysis on exhaust emissions has been done.
The insulated semi-infinite cylinder with heat supplied uniformly through a coaxial contact area is an important unit cell in the theory of contact resistance. In the past, several investigators have examined the problem of the thermal constriction resistance of a circular contact area on an insulated semi-infinite, coaxial circular cylinder. Recently the case of a circular contact on a square cylinder has been investigated. By using a new approximate technique as well as double-infinite series solution to Laplace's equation in Cartesian coordinates, the case of a square contact area on a square cylinder can also be solved. Previously the nondimensionalization technique employed for the results has varied with different configurations and researchers. However, the similarity of these three configurations in terms of their integrated parameter, the thermal constriction resistance, will only be seen when nondimensionalization is made using a characteristic dimension that best describes these geometries.
An infinite interface of randomly distributed contacts is modeled as a finite square region with randomly placed contacts
inside it. The contacts outside the region are treated as continuum of contacts. The continuum approximation allows for an
interaction between the contacts within the square and those outside it. An analytical solution is obtained for the temperature
field, and the contact resistance is analyzed for randomness effects. This is the first such analytical model developed to
study random distribution of contacts. The result shows an excellent agreement when tested against the the available analytical
solution for the case of periodic arrangement of contacts. For the random case, the resistance is observed to be a strong
function of the area fraction of contact.
A method is suggested for the calculation of the temperature of large semitransparent particles of a material of low thermal conductivity during their motion in a gas jet. The temperature profile in a particle is calculated with due regard for thermal conductivity and thermal radiation. The results of calculations performed for typical parameters of a plasma jet demonstrate that the temperature at the center of an oxide particle 40–60 m in diameter in the zone of heating may be 1500–2500 K lower than the temperature at the particle surface. Here, the error of prediction of the volume-average particle temperature in isothermal approximation reaches 350 K. The possibility of determining the volume-average temperature of particles by their thermal radiation is studied as applied to the problem of optical diagnostics of two-phase jets. It is demonstrated that the color temperature being measured may differ from the volume-average particle temperature by as much as 300 K. The possibility is discussed of experimentally determining the temperature dependence of the index of absorption, which would enable one to considerably increase the accuracy of the obtained values of the volume-average particle temperature.
The applicability of radiation transfer theory for calculations of the thermal radiation emitted by spherical particle of a semitransparent material, and in particular the determination of radial heat generation profiles, is analyzed. For homogeneous isothermal particles, a comparison with the exact solution based on the Mie theory shows that the radiation transfer calculations are sufficiently accurate for diffraction parameter of the particle of x ≥ 20. Numerical examples for large particles illustrate the transition from conditions of dominant radiation of the central region of the particle to conditions of the surface layer emission. A new differential approximation for radiation transfer in a refracting particle is proposed. This approximation called MDP0 (modified DP0) is much simpler than the radiation transfer equation. Using MDP0, we have a chance to consider radiation–conduction interaction in nonisothermal particles without great computational efforts.
- M G Cooper
- B B Mikic
Cooper, M. G., Mikic, B. B., and Yovanovich, M. M., 1969, ''Thermal Contact
Conductance,'' Int. J. Heat Mass Transf., 17, pp. 205-214.
Inclusion of Nonisothermality of Particles in the Calculations and Diagnostics of Two-Phase Jets Used for Spray Coating
- L A Dombrovsky
- M B Ignatiev