7 Dilute Dispersed-Phase and Interfacial-Area Model

NeuralFlow contains a dilute Eulerian dispersed-phase model coupled to the carrier equations through momentum and heat-transfer source terms. The model is intended for droplets, particles, or agglomerates whose local volume fraction is small enough that carrier-volume displacement is neglected. This assumption is distinct from the numerical packing protection discussed below: a local friction pressure and a conservative packing limiter regularize the pressureless dispersed subsystem, but they do not turn the formulation into a dense two-fluid model.

A central feature of the NeuralFlow formulation is that particle size can be either prescribed (monodisperse mode) or evolved through a one-group interfacial-area transport equation (IATE). When IATE is active, breakup and coalescence modify the interfacial area concentration and therefore the representative diameter. The resulting size then feeds back into drag, heat transfer, response times, and the dispersed-phase numerical flux.

7.1 Dispersed-phase state and geometric identities

The primary dispersed mass variable is the dispersed mass concentration (7.1) where is the dispersed volume fraction and is the intrinsic density of the condensed material. Consequently, (7.2) and the packing constraint can equivalently be written

The primitive dispersed state is schematically with omitted when IATE is disabled. Here is the interfacial area concentration, with units .

For a monodisperse population of spherical particles, (7.3) so that the representative diameter reconstructed by the NeuralFlow IATE path is (7.4) Equation (7.4) is the important kinematic closure: carries dispersed mass, whereas carries surface-area information. At fixed dispersed mass, breakup increases and decreases ; coalescence decreases and increases .

The total surface area density is , but the projected frontal area density entering a conventional spherical drag law is (7.5) This distinction is important: using directly as projected area would overestimate the spherical drag area by a factor of four.

7.2 Mass, momentum, and thermal transport

The dispersed mass balance is (7.6) The momentum balance with packing/friction pressure is (7.7) Here is the force on the dispersed phase from the carrier. The additional term is not a thermodynamic pressure; it is a packing-activated regularization of the pressureless particulate subsystem.

The dispersed thermal primitive variable is . A compact transported balance is , with . Particle material properties therefore participate directly in the coupled thermal response; particle temperature is not merely a post-processing field.

Momentum exchange changes dispersed kinetic energy even if the stored particle thermal quantity is not a carrier-style total-energy variable. Therefore equal-and-opposite momentum coupling alone does not guarantee combined carrier/dispersed energy conservation. The presently current drag-work partition is identified as an open formulation issue in Chapter 20.

7.3 Drag coupling and particle response time

Define the slip velocity For spherical particles, the force per mixture volume associated with a conventional drag coefficient is (7.9) The particle Reynolds number is (7.10) A Schiller–Naumann-type correlation is available for dilute spherical particles (Schiller and Naumann 1935). For example, in its usual moderate-Reynolds-number form, subject to the Reynolds-range and high- branch used by the selected NeuralFlow model option.

A useful relaxation scale follows by comparing the drag force with dispersed momentum. In the Stokes limit, (7.11) Thus breakup can sharply decrease the local momentum-response time through its dependence, while coalescence has the opposite effect.

7.4 Interphase heat transfer

The convective carrier-to-particle heat-transfer source is represented by (7.12) For isolated spheres, a Ranz–Marshall form is (7.13) (Ranz and Marshall 1952). The corresponding thermal-response scale is approximately (7.14) Hence the IATE feedback is twofold: changes the available transfer area directly and, through Eq. (7.4), changes the Reynolds/Nusselt numbers and the response-time scales.

7.5 Pressureless degeneracy and packing regularization

If and sources are omitted, the one-dimensional dispersed mass–momentum subsystem is Both characteristic speeds collapse to , so the system is only weakly hyperbolic. NeuralFlow regularizes this degeneracy only when the local concentration enters a near-packing regime.

Let be the friction-pressure activation threshold. The regularization is inactive in the dilute region, (7.15) and above the threshold uses a power-law form (7.16) The effective coefficient is velocity-gated, (7.17) which prevents the regularization from generating an arbitrarily large pseudo-acoustic speed in slowly moving concentrated material.

The corresponding acoustic-like speed is (7.18) with the velocity-gated bound (7.19) The face-normal characteristic speeds then become (7.20) so strict hyperbolicity is restored wherever . The friction pressure supplies a finite signal speed; the separate receiver-side packing limiter in Chapter 8 enforces the hard bound .

7.6 One-group interfacial-area transport equation

The NeuralFlow one-group IATE is (7.21) with (7.22) Here is the random-collision or turbulence-induced coalescence contribution, is acceleration- or Rayleigh–Taylor-type breakup, is turbulent-impact breakup, and is a gravity/Eotvos-type breakup contribution. The one-group concept follows the general IATE framework (Ishii, Kim, and Uhle 2002; Morel, Goreaud, and Delhaye 1999), while the selected source closures are NeuralFlow model options.

Since , all IATE source terms have dimensions (7.23) This dimensional check is useful when implementing or modifying breakup/coalescence closures.

7.7 Turbulence quantities used by IATE closures

The turbulent-impact and coalescence options use the carrier GE - turbulence state when that model is active. The local dissipation-rate proxy is , with for these closures. The Kolmogorov scaling then provides the turbulence velocity scale at particle size .

Turbulence-driven coalescence and breakup require the dispersed model, interfacial-area transport and the GE-RANS turbulence variables. If those prerequisites are not active, the corresponding contribution is zero. This means checking a breakup/coalescence box without the required turbulence/IATE setup does not create the intended physical source.

7.8 Turbulence-induced coalescence

Coalescence merges dispersed entities and therefore reduces total area, The NeuralFlow Luo1993 option is formulated as a collision-frequency process multiplied by a coalescence efficiency (Luo 1993). It is useful to write the source structurally as (7.25) where is the collision-frequency/area-loss scale, collects model constants, and is the probability that a collision results in coalescence.

The physical competition is between turbulent collision/contact and the surface-tension-controlled resistance to merging. Therefore increasing turbulent agitation generally raises the collision frequency, while increasing surface tension tends to suppress successful coalescence. Because Eq. (7.4) couples back to , a negative increases the representative diameter at approximately fixed dispersed mass.

The selectable turbulence-induced coalescence option is coupled to , , dispersed mass concentration, interfacial area, carrier density, particle material density and surface tension. Because the coupling is implicit, changing the coalescence constants can alter both the predicted size evolution and nonlinear convergence behavior.

7.9 Turbulent-impact breakup

The turbulent-impact breakup option uses the dissipation proxy of Eq. (7.24). Its breakup efficiency is

The used in NeuralFlow turbulent-impact area source has the dimensional structure (7.27) where denotes the model coefficient used by the selected closure and denotes the safeguarded packing-related factor in the formulation. The source path uses a protected remaining-packing measure based on (7.28) so that the algebra remains finite as the packing limit is approached. The precise algebraic placement of this safeguarded factor is retained as an formulation detail rather than silently replacing it by an unsafeguarded analytical expression.

Equation (7.27) has the required units: contributes , contributes , and contributes , giving before dimensionless coefficients and packing factors.

The turbulent-impact source is positive when active, and therefore increases and reduces .

7.10 Acceleration/Rayleigh–Taylor-type breakup

Strong relative acceleration can destabilize a dense particle or droplet against the surrounding gas. NeuralFlow includes an acceleration-driven/Rayleigh–Taylor-type branch in which the local diameter is compared with a critical stable length scale set by surface tension, density contrast, and destabilizing acceleration. A representative scaling is (7.29) where is the relevant relative/body acceleration magnitude used by the closure.

The acceleration-driven breakup source is activated smoothly around its critical-size condition rather than with a discontinuous switch. When the representative diameter exceeds the stable diameter predicted by the model, the source is positive, increasing interfacial area and reducing the representative diameter. The smooth activation improves nonlinear behavior close to the threshold.

Equation (7.29) documents the physical scaling rather than asserting an unverified coefficient. NeuralFlow model constants and the exact active acceleration definition must be kept synchronized with the model formulation when this closure is modified.

7.11 Gravity/Eotvos-type breakup

For body-force-dominated deformation, NeuralFlow includes an Eotvos-type breakup branch. The controlling dimensionless group is (7.31) where is the acceleration magnitude used by the model. Rearranging a critical value gives the corresponding stable-diameter scale (7.32) The source drives the area toward the value corresponding to a smaller stable diameter when the Eotvos criterion is exceeded. In one-group form this can be expressed schematically as (7.33) with denoting the model activation. The sign is positive in the breakup regime because when the current particles are larger than the stable size.

7.12 Competition between coalescence and breakup

The net source can be positive or negative, (7.34) A useful local size-evolution identity follows from Eq. (7.4). Neglecting material-density variation for clarity, (7.35) For a homogeneous parcel with no net dispersed-mass source, an IATE source therefore gives approximately (7.36) This makes the physical interpretation immediate: breakup () decreases diameter, while coalescence () increases it.

7.13 Packing admissibility and source protection

The friction pressure and the packing limiter have different roles. The friction pressure alters characteristic propagation in concentrated cells; the transport limiter enforces the hard state bound. Source terms must also preserve admissibility and numerical regularity. In particular,

• and must remain positive enough for Eq. (7.4) to be meaningful;

• divisions by , , or remaining packing margin must use the formulation safeguards;

• source switches should be differentiable wherever practical because their derivatives enter the implicit Jacobian; and

• breakup/coalescence changes area/size, not dispersed mass, unless a separate physical mass-transfer model is explicitly active.

7.14 7.14 Implicit coupling

Particle convection, friction pressure, drag, heat transfer, gravity, IATE transport and breakup/coalescence participate in the same coupled nonlinear system as the carrier equations. Consequently, a change in particle concentration, interfacial area or particle temperature can influence drag/heat-transfer coefficients and can also change carrier-flow convergence through the interphase coupling.

For a local IATE source , (7.37) The resulting cross-coupling is particularly important for stiff drag/heat-transfer regimes and for breakup models in which turbulence variables strongly control the source rate.

The formulation remains a dilute dispersed-phase model. It does not include carrier volume displacement, dense-phase thermodynamic pressure, a granular-temperature equation, or a resolved population-balance distribution unless those models are added separately.

7.15 Current selectable KH/Reitz branch

When the GUI option Particle Breakup (Reitz 1999) is selected, NeuralFlow uses the acceleration-driven Rayleigh–Taylor-style interfacial-area formulation described in this section. Users should therefore interpret this selector by the formulation given here rather than as a standalone wavelength-only Kelvin–Helmholtz breakup model. The aerodynamic acceleration uses the spherical projected-area density .

It then forms

and smoothly activates breakup when exceeds . The growth-rate/time-scale chain is

The raw area-source rate is proportional to

multiplied by the smooth activation and the cell volume before insertion into the interfacial-area row. This manual therefore names the selector and the actual mathematical source separately to avoid implying a different closure than the one executed by the source.