Kenneth Harstad

Evaluation of equilibrium and non-equilibrium evaporation models for many-droplet gas-liquid flow simulations

A variety of liquid droplet evaporation models, including both classical equilibrium and non-equilibrium Langmuir–Knudsen formulations, are evaluated through comparisons with experiments with particular emphasis on computationally efficient procedures for gas–liquid flow simulations. The models considered are those used in droplet laden flow calculations such as direct numerical simulations for which large numbers of individual (isolated) droplet solutions are obtained. Diameter and temperature evolution predictions are made for single-component droplets of benzene, decane, heptane, hexane and water with relatively large initial sizes ∼1 mm vaporizing in convective air flows. All of the models perform nearly identically for low evaporation rates at gas temperatures significantly lower than the boiling temperature. For gas temperatures at and above the boiling point, large deviations are found between the various model predictions. The simulated results reveal that non-equilibrium effects become significant when the initial droplet diameter is <50 μm and that these effects are enhanced with increasing slip velocity. It is additionally observed that constant properties can be used throughout each simulation if both the gas and vapor values are calculated at either the wet-bulb or boiling temperature. The models based on the Langmuir–Knudsen law and a corrected (for evaporation effects) analytical heat transfer expression derived from the quasi-steady gas phase assumption are shown to agree most favorably with a wide variety of experimental results. Since the experimental droplet sizes are all much larger than the limit for non-equilibrium effects to be important, for these conditions the most crucial aspect of the current Langmuir–Knudsen models is the corrected analytical form for the heat transfer expression as compared to empirical relations used in the remaining models.

Direct numerical simulations of supercritical fluid mixing layers applied to heptane–nitrogen

Direct numerical simulations (DNS) are conducted of a model hydrocarbon{nitrogen mixing layer under supercritical conditions. The temporally developing mixing layer conguration is studied using heptane and nitrogen supercritical fluid streams at a pressure of 60 atm as a model system related to practical hydrocarbon-fuel/air systems. An entirely self-consistent cubic Peng{Robinson equation of state is used to describe all thermodynamic mixture variables, including the pressure, internal energy, enthalpy, heat capacity, and speed of sound along with additional terms associated with the generalized heat and mass transport vectors. The Peng{Robinson formulation is based on pure-species reference states accurate to better than 1% relative error through comparisons with highly accurate state equations over the range of variables used in this study (6006 T 6 1100 K, 406 p6 80 atm) and is augmented by an accurate curve t to the internal energy so as not to require iterative solutions. The DNS results of two-dimensional and three-dimensional layers elucidate the unique thermodynamic and mixing features associated with supercritical conditions. Departures from the perfect gas and ideal mixture conditions are quantied by the compression factor and by the mass diusion factor, both of which show reductions from the unity value. It is found that the qualitative aspects of the mixing layer may be dierent according to the specication of the thermal diusion factors whose value is generally unknown, and the reason for this dierence is identied by examining the second-order statistics: the constant Bearman{Kirkwood (BK) thermal diusion factor excites fluctuations that the constant Irwing{Kirkwood (IK) one does not, and thus enhances overall mixing. Combined with the eect of the mass diusion factor, constant positive large BK thermal diusion factors retard diusional mixing, whereas constant moderate IK factors tend to promote diusional mixing. Constant positive BK thermal diusion factors also tend to maintain density gradients, with resulting greater shear and vorticity. These conclusions about IK and BK thermal diusion factors are speciespair dependent, and therefore are not necessarily universal. Increasing the temperature of the lower stream to approach that of the higher stream results in increased layer growth as measured by the momentum thickness. The three-dimensional mixing layer exhibits slow formation of turbulent small scales, and transition to turbulence does not occur even for a relatively long non-dimensional time when compared to a previous, atmospheric conditions study. The primary reason for this delay is the initial density stratication of the flow, while the formation of strong density gradient regions both in the braid and between-the-braid planes may constitute a secondary reason for the hindering of transition through damping of emerging turbulent eddies.

An All-Pressure Fluid Drop Model Applied to a Binary Mixture: Heptane in Nitrogen

The differences between subcritical liquid drop and supercritical fluid drop behavior are shown to be a direct consequence of the length scales near the fluid drop boundary. Under subcritical, evaporative high emission rate conditions, a film layer is present in the inner part of the drop surface which contributes to the unique determination of the boundary conditions; it is this film layer in conjunction with evaporation which gives to the solution its convective–diffusive character. In contrast, under supercritical conditions the boundary conditions contain a degree of arbitrariness due to the absence of a physical surface, and the solution has then a purely diffusive character. Results from simulations of a free fluid drop under no-gravity conditions are compared to microgravity experimental data from suspended, large drop experiments at high, low and intermediary temperatures and in a range of pressures encompassing the sub- and supercritical regime. Despite the difference between the conditions of the simulations and the experiments, the time rate of variation of the drop diameter square is remarkably well predicted in the linear curve regime. Consistent with the optical measurements, in the simulations the drop diameter is determined from the location of the maximum density gradient. Detailed time-wise comparisons between simulations and data show that this location is very well predicted at 0.1 MPa. As the pressure increases, the data and simulations agreement becomes good to fair, and the possible reasons for this discrepancy are discussed. Simulations are further conducted for a small drop, such as that encountered in practical applications, over a wide range of specified, constant far field pressures. Additionally, a transient pressure simulation crossing the critical point is also conducted. Results from these simulations are analyzed and major differences between the sub- and supercritical behavior are explained. In particular, it is shown that the classical calculation of the Lewis number gives erroneous results at supercritical conditions, and that an effective Lewis number previously defined gives correct estimates of the length scales for heat and mass transfer at all pressures.

Modelling of subgrid-scale phenomena in supercritical transitional mixing layers: an a priori study

A database of transitional direct numerical simulation (DNS) realizations of a supercritical mixing layer is analysed for understanding small-scale behaviour and examining subgrid-scale (SGS) models duplicating that behaviour. Initially, the mixing layer contains a single chemical species in each of the two streams, and a perturbation promotes roll-up and a double pairing of the four spanwise vortices initially present. The database encompasses three combinations of chemical species, several perturbation wavelengths and amplitudes, and several initial Reynolds numbers specifically chosen for the sole purpose of achieving transition. The DNS equations are the Navier-Stokes, total energy and species equations coupled to a real-gas equation of state; the fluxes of species and heat include the Soret and Dufour effects. The large-eddy simulation (LES) equations are derived from the DNS ones through filtering. Compared to the DNS equations, two types of additional terms are identified in the LES equations: SGS fluxes and other terms for which either assumptions or models are necessary. The magnitude of all terms in the LES conservation equations is analysed on the DNS database, with special attention to terms that could possibly be neglected. It is shown that in contrast to atmospheric-pressure gaseous flows, there are two new terms that must be modelled: one in each of the momentum and the energy equations. These new terms can be thought to result from the filtering of the nonlinear equation of state, and are associated with regions of high density-gradient magnitude both found in DNS and observed experimentally in fully turbulent high-pressure flows. A model is derived for the momentum-equation additional term that performs well at small filter size but deteriorates as the filter size increases, highlighting the necessity of ensuring appropriate grid resolution in LES. Modelling approaches for the energy-equation additional term are proposed, all of which may be too computationally intensive in LES. Several SGS flux models are tested on an a priori basis. The Smagorinsky (SM) model has a poor correlation with the data, while the gradient (GR) and scale-similarity (SS) models have high correlations. Calibrated model coefficients for the GR and SS models yield good agreement with the SGS fluxes, although statistically, the coefficients are not valid over all realizations. The GR model is also tested for the variances entering the calculation of the new terms in the momentum and energy equations; high correlations are obtained, although the calibrated coefficients are not statistically significant over the entire database at fixed filter size. As a manifestation of the small-scale supercritical mixing peculiarities, both scalar-dissipation visualizations and the scalar-dissipation probability density functions (PDF) are examined. The PDF is shown to exhibit minor peaks, with particular significance for those at larger scalar dissipation values than the mean, thus significantly departing from the Gaussian behaviour.

Direct numerical simulations of O2/H2 temporal mixing layers under supercritical conditions

Direct numerical simulations of a supercritical oxygen/hydrogen temporal three-dimensional mixing layer are conducted to explore the features of high-pressure transitional mixing behavior. The conservation equations are formulated according to fluctuation–dissipation theory and are coupled to a modified Peng–Robinson equation of state. The boundary conditions are periodic in the streamwise and spanwise directions and of nonreflecting outflow type in the cross-stream direction. Simulations are conducted with initial Reynolds numbers of 6 x 10^2 and 7.5 x 10^2, initial pressure of 100 atm, and temperatures of 400 K in the O_2 and 600 K in the H_2 stream. Each simulation encompasses the rollup and pairing of four initial spanwise vortices into a single vortex. The layer eventually exhibits distorted regions of high density-gradient-magnitude similar to the experimentally observed wisps of fluid at the boundary of supercritical jets. Analysis of the data reveals that the higher-Reynolds-number layer reaches transition, whereas the other one does not. The transitional layer is analyzed to elucidate its characteristics.

Statistical model of multicomponent-fuel drop evaporation for many-drop flow simulations

A statistical formulation is developed describing the composition in an evaporating multicomponent-fuel liquid drop and in the gas phase surrounding it. When a complementary discrete-component model is used, it is shown that, when drops are immersed in a carrier gas containing fuel vapor, condensation of species onto the drop results in the development of a minor peak in the liquid composition probability distribution function (PDF). This peak leads to a PDF shape that can be viewed as a combination of two gamma PDFs, which is determined by five parameters. A model is developed for calculating the parameters of the two combined gamma PDFs. Extensive tests of the model for both diesel and gasoline show that the PDF results replicate accurately the discrete model predictions. Most important, the mean and variance of the composition at the drop surface are in excellent agreement with the discrete model. Results from the model show that although the second peak is minor for the liquid PDF, its corresponding peak for the vapor distribution at the drop surface has a comparable magnitude to and sometimes exceeds that corresponding to the first peak. Four-parameter models are also exercised, and it is shown that they are unable to capture the physics of the problem.

Evaluation of commonly used assumptions for isolated and cluster heptane drops in nitrogen at all pressures

A study is performed to assess commonly used assumptions in the modeling of drop behavior in moderate to high temperature surroundings and at all pressures. The model employed for this evaluation has been previously validated for isolated drops by using microgravity data, and is very general: it contains Soret and Dufour effects, does not assume mass transfer quasi-steadiness at the drop boundary, or necessarily the existence of a drop surface (i.e., phase discontinuity). Moreover, the numerical simulations are performed with accurate equations of state and transport properties over a wide range of thermodynamic variables. Consistent with low pressure conditions, the drop boundary is identified a posteriori of the calculations with the location of the largest density change. Simulations are here performed for isolated drops, and for monodisperse as well as binary size drop clusters. The results show that at locations arbitrarily near the boundary, the drop does not reach the mixture critical point within the wide range of conditions investigated (far-field temperatures of 470–1000 K and pressures ranging from 0.1 to 5 MPa). However, the state arbitrarily near the boundary is closer to the critical condition for smaller drops in a cluster than for the larger drops. Evaluations of the effect of the relaxation time at the drop boundary show that quasi-steadiness of the mass transfer prevails for drops of radius as small as 2 × 10^(−3) cm. Finally, the diameter squared exhibits a linear time variation only at atmospheric pressure. At all other pressures investigated (1–5 MPa), the diameter squared displays a negative curvature with time which never becomes linear. In agreement with existing experimental data, the drop lifetime increases monotonically with pressure at low far field temperatures (470 K), but exhibits a maximum as a function of pressure at high temperatures (1000 K). On an appropriate scale, the slope of the diameter squared versus time is shown to be independent of the drop size at all pressures.

Evaporation, ignition and combustion of nondilute clusters of drops

A theory of evaporation, ignition, and burning of moderately dense spherical drop clusters has been developed. The theory takes into account burning of premixed air and fuel internal to the cluster at ignition and subsequent burning of fuel emitted from the cluster by a flame sheet surrounding it. The model considers interdrop interaction, momentum exchange between drops and gas, and turbulent exchange processes between the cluster and its surroundings. Calculations are performed for varying initial air-to-fuel-mass ratios, initial cluster radii, ambient gas temperatures and initial drop temperatures. Results are presented for ratios of fuel mass burned to fuel mass lost from the cluster between drop ignition and drop disappearance, fuel burned fractions at ignition and at the moment of drop disappearance, and jump conditions at ignition.

The D2 variation for isolated LOX drops and polydisperse clusters in hydrogen at high temperature and pressures

A study of the d^2 variation for isolated fluid drops and for fluid drops belonging to polydisperse clusters has been conducted at a high temperature and elevated pressures. The mathematical formulation is based on a previously validated model of subcritical/supercritical isolated fluid drop behavior. Coupled with the isolated drop equations, a set of conservation equations has been developed to describe the global cluster behavior. All these equations are based on the general transport matrix including Soret and Dufour terms and they are consistent with nonequilibrium thermodynamics and at low pressure with kinetic theory. Moreover, the model also accounts for real gas effects through accurate equations of state and for correct values of the transport properties in the high pressure, high temperature regime. The model has been first exercised for isolated LOX drops in H_2 at pressures ranging from 1.5 MPa (subcritical pressure for O_2) to 20 MPa (supercritical pressure for O_2). The results show that while at subcritical pressures the d^2 variation is nearly linear, with increasing pressure it departs considerably from the linear behavior; the largest departure occurs in the vicinity of the oxygen critical point. The slope of d^2(t) was fitted using both a constant and a linear fit, and it was shown that the linear fit provides a better alternative for correlation purposes. Simulations were also conducted for clusters of LOX drops in H_2 in the range 6 to 40 MPa (reduced pressures of 1.2–8 with respect to pure O_2). Parametric studies of the effect of the thermal diffusion factor value reveal that it is minor at 10 MPa and moderate at 40 MPa, and that although the Soret term is dominated by the Fick, Dufour, and Fourier terms, it is not negligible. The influence of a cluster Nusselt number is also shown to be relatively small in the range 10^3 to 10^4, consistent with the supercritical behavior being essentially a diffusive one. All of the results show a nonlinear d^2 variation with curves having a positive curvature independent of the values of the thermal diffusion factor, the Nusselt number or the LOX/H_2 mass ratio. The approximation of a binary size cluster containing relatively a much larger number of small drops by a monodisperse cluster with a drop size based upon the surface average of the drops in the polydisperse cluster yields a good evaluation of the thermodynamic quantities in the interstitial drop region but an underestimate of the lifetime of the drops in the cluster.

Behavior of a polydisperse cluster of interacting drops evaporating in an inviscid vortex

The dynamics and evaporation of polydisperse collections of liquid drops in an axisymmetric, infinite, cylindrical vortex are described using a statistical model. This model describes both the dense regime where inter-particle effects are important and the dilute regime. The initial size distribution is partitioned into size classes and each initial size-class is followed dynamically and thermodynamically using a class-defined, drop-frame coordinate system. Each initial-size-class develops a continuum of sizes as drops centrifuge towards hotter surroundings and evaporate. A separate coordinate system tracks the gas phase. Because larger drops experience larger centrifugal force, they approach the hotter gas faster. However, for appropriate liquid heating times, the large drops might evaporate at a faster rate, and so the size-differentiated centrifugation previously observed and calculated for cold flow situations does not occur. Instead, a radially peaked drop size distribution is developed in the gas vortex. The centrifugal motion forms a drop-free inner vortex core bound by a cylindrical shell containing all the drops. This shell of gas and drops is called the drop cluster. Numerical calculations show that more parameters control dense clusters than dilute clusters; examples of these parametric relations include: (i the gas vortex, whereas drop size distribution controls the outer region; and (ii increases the maximum mass fraction of the evaporated compound and enhances penetration of the evaporated compound into the surroundings. Except for dilute clusters, the assumption of uniform drop number distribution in the cluster is found to be inappropriate. Instead, the drop size distribution always becomes non-uniform even if the initial size distribution is monodisperse and the initial drop number distribution is uniform. This development of non-uniformity is caused by drops at the cluster peripheries preventing heat conduction/convection to drops in the central cluster.