An equation set for non-equilibrium two phase flow, and an analysis of some aspects of choking, acoustic propagation, and losses in low pressure wet steam

Jackson, R.J.; Davidson, B.J.

doi:10.1016/0301-9322(83)90014-9

Cited by 24 publications

(6 citation statements)

References 1 publication

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“…solid particles) by simply inserting dm d /dt = 0. The equations presented in this section for the Eulerian-Lagrangian system are in agreement with the analogous equations presented in the Eulerian-Eulerian framework by Jackson & Davidson (1983).…”

Section: Coupling Termssupporting

confidence: 81%

Droplet–turbulence interactions in low-Mach-number homogeneous shear two-phase flows

Mashayek

1998

J. Fluid Mech.

115

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Several important issues pertaining to dispersion and polydispersity of droplets in turbulent flows are investigated via direct numerical simulation (DNS). The carrier phase is considered in the Eulerian context, the dispersed phase is tracked in the Lagrangian frame and the interactions between the phases are taken into account in a realistic two-way (coupled) formulation. The resulting scheme is applied for extensive DNS of low-Mach-number, homogeneous shear turbulent flows laden with droplets. Several cases with one- and two-way couplings are considered for both non-evaporating and evaporating droplets. The effects of the mass loading ratio, the droplet time constant, and thermodynamic parameters, such as the droplet specific heat, the droplet latent heat of evaporation, and the boiling temperature, on the turbulence and the droplets are investigated. The effects of the initial droplet temperature and the initial vapour mass fraction in the carrier phase are also studied. The gravity effects are not considered as the numerical methodology is only applicable in the absence of gravity. The evolution of the turbulence kinetic energy and the mean internal energy of both phases is studied by analysing various terms in their transport equations. The results for the non-evaporating droplets show that the presence of the droplets decreases the turbulence kinetic energy of the carrier phase while increasing the level of anisotropy of the flow. The droplet streamwise velocity variance is larger than that of the fluid, and the ratio of the two increases with the increase of the droplet time constant. Evaporation increases both the turbulence kinetic energy and the mean internal energy of the carrier phase by mass transfer. In general, evaporation is controlled by the vapour mass fraction gradient around the droplet when the initial temperature difference between the phases is negligible. In cases with small initial droplet temperature, on the other hand, the convective heat transfer is more important in the evaporation process. At long times, the evaporation rate approaches asymptotic values depending on the values of various parameters. It is shown that the evaporation rate is larger for droplets residing in high-strain-rate regions of the flow, mainly due to larger droplet Reynolds numbers in these regions. For both the evaporating and the non-evaporating droplets, the root mean square (r.m.s.) of the temperature fluctuations of both phases becomes independent of the initial droplet temperature at long times. Some issues relevant to modelling of turbulent flows laden with droplets are also discussed.

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Section: Coupling Termssupporting

confidence: 81%

Droplet–turbulence interactions in low-Mach-number homogeneous shear two-phase flows

Mashayek

1998

J. Fluid Mech.

115

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“…The local summations in (18)-(20) are necessarily grid-dependent functions; the summations are over all droplets (subscript α indicates the individual droplet variables; no summation over Greek indices) residing within a local numerical discretization volume (∆V α ) and employ a geometrical weighting factor, w α (discussed below). In the above form, the phase-coupling terms are in agreement with the derivations of Jackson & Davidson (1983) and Young (1995) for Eulerian dispersed-phase flow descriptions, and also with those of Mashayek (1998a,b) which were used to simulate the Lagrangian evolution of evaporating droplets in homogeneous turbulence. Note that neither of the first two citations addresses the evaluation of h V ,s , whereas Mashayek only considers flows for which the liquid and vapour heat capacities are equal and constant, in which case the correct evaluation of the vapour enthalpy is greatly simplified.…”

Section: Individual Droplet Conservation Equationssupporting

confidence: 75%

Direct numerical simulation of a confined three-dimensional gas mixing layer with one evaporating hydrocarbon-droplet-laden stream

Miller¹,

1999

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Direct numerical simulations are performed of a confined three-dimensional, temporally developing, initially isothermal gas mixing layer with one stream laden with as many as 7.3×10 5 evaporating hydrocarbon droplets, at moderate gas temperature and subsonic Mach number. Complete two-way phase couplings of mass, momentum and energy are incorporated which are based on a thermodynamically self-consistent specification of the vapour enthalpy, internal energy and latent heat of vaporization. Effects of the initial liquid mass loading ratio (ML), initial Stokes number (St 0 ), initial droplet temperature and flow three-dimensionality on the mixing layer growth and development are discussed. The dominant parameter governing flow modulation is found to be the liquid mass loading ratio. Variations in the initial Stokes number over the range 0.5 6 St 0 6 2.0 do not cause significant modulations of either first-or second-order gas phase statistics. The mixing layer growth rate and kinetic energy are increasingly attenuated for increasing liquid loadings in the range 0 6 ML 6 0.35. The laden stream becomes saturated before evaporation is completed for all but the smallest liquid loadings owing to: (i) latent heat effects which reduce the gas temperature, and (ii) build up of the evaporated vapour mass fraction. However, droplets continue to be entrained into the layer where they evaporate owing to contact with the relatively higher-temperature vapour-free gas stream. The droplets within the layer are observed to be centrifuged out of high-vorticity regions and to migrate towards high-strain regions of the flow. This results in the formation of concentration streaks in spanwise braid regions which are wrapped around the periphery of secondary streamwise vortices. Persistent regions of positive and negative slip velocity and slip temperature are identified. The velocity component variances in both the streamwise and spanwise directions are found to be larger for the droplets than for the gas phase on the unladen stream side of the layer; however, the cross-stream velocity and temperature variances are larger for the gas. Finally, both the mean streamwise gas velocity and droplet number density profiles are observed to coincide for all ML when the cross-stream coordinate is normalized by the instantaneous vorticity thickness; however, first-order thermodynamic profiles do not coincide.

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“…This was because he did not define the relaxation time via an archetypal equation like ( 2 1 ) . The expression given by Jackson & Davidson (1983) is also incorrect for the same reason.…”

Section: At % (T-t)mentioning

confidence: 99%

Normal shock-wave structure in two-phase vapour-droplet flows

Young

Guha

1991

JFM

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A study of the structure of stationary, fully and partly dispersed, normal shock waves in steady vapour–droplet, two-phase flow is presented. Pure substances only are considered, but, unlike most previous work, the droplet population is allowed to be polydispersed. It is shown how the effects of thermal relaxation for such a mixture can be elegantly incorporated into the analysis.Three types of fully dispersed wave are identified. Type I waves are dominated by thermal relaxation and an approximate analytical solution is presented which gives results in close agreement with accurate numerical solutions of the governing equations. The analysis predicts some unexpected behaviour of the thermodynamic variables and demonstrates the correct scaling parameters for such flows. An approximate analysis is also presented for Type II waves, dominated by both velocity and thermal relaxation. Type III waves, where all three relaxation processes are important, are of little practical significance and are only briefly discussed. Partly dispersed waves are also considered and the results of a numerical simulation of the relaxation zone are presented. A linearized solution of this problem is possible but, unlike other relaxing gas flows, does not give good agreement with the more exact numerical calculations.The apparent discontinuity in the speed of sound in a vapour–droplet mixture as the wetness fraction tends to zero has been responsible for some confusion in the literature. This problem is reconsidered and it is shown that the transition from the two-phase equilibrium to the single-phase frozen shock wave speed is continuous.

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An equation set for non-equilibrium two phase flow, and an analysis of some aspects of choking, acoustic propagation, and losses in low pressure wet steam

Cited by 24 publications

References 1 publication

Droplet–turbulence interactions in low-Mach-number homogeneous shear two-phase flows

Droplet–turbulence interactions in low-Mach-number homogeneous shear two-phase flows

Direct numerical simulation of a confined three-dimensional gas mixing layer with one evaporating hydrocarbon-droplet-laden stream

Normal shock-wave structure in two-phase vapour-droplet flows

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