7 Dilute Dispersed-Phase and Interfacial-Area Model

CMPS 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 CMPS implementation 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 \[\rho_d=\alpha\,\rho_{p,m}(T_d), \](7.1) where \(\alpha\) is the dispersed volume fraction and \(\rho_{p,m}\) is the intrinsic density of the condensed material. Consequently, \[\alpha=\frac{\rho_d}{\rho_{p,m}(T_d)}, \qquad 0\le\alpha\le\alpha_{\max}, \](7.2) and the packing constraint can equivalently be written \[0\le \rho_d\le\alpha_{\max}\rho_{p,m}(T_d).\]

The primitive dispersed state is schematically \[\mathbf{q}_d=(\rho_d,\mathbf{V}_d,T_d,a_i),\] with \(a_i\) omitted when IATE is disabled. Here \(a_i\) is the interfacial area concentration, with units \(\mathrm{m}^{-1}\).

For a monodisperse population of spherical particles, \[a_i=\frac{6\alpha}{d_d}, \](7.3) so that the representative diameter reconstructed by the CMPS IATE path is \[d_d=\frac{6\rho_d}{\rho_{p,m}a_i}. \](7.4) Equation (7.4) is the important kinematic closure: \(\rho_d\) carries dispersed mass, whereas \(a_i\) carries surface-area information. At fixed dispersed mass, breakup increases \(a_i\) and decreases \(d_d\); coalescence decreases \(a_i\) and increases \(d_d\).

The total surface area density is \(a_i\), but the projected frontal area density entering a conventional spherical drag law is \[A_{\mathrm{proj}}=\frac{a_i}{4}. \](7.5) This distinction is important: using \(a_i\) 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 \[\frac{\partial\rho_d}{\partial t} +\nabla\cdot(\rho_d\mathbf{V}_d)=0. \](7.6) The momentum balance with packing/friction pressure is \[\frac{\partial(\rho_d\mathbf{V}_d)}{\partial t} +\nabla\cdot \left(\rho_d\mathbf{V}_d\otimes\mathbf{V}_d+p_{fr}\mathbf{I}\right) =\mathbf{F}_{dg}+\rho_d\mathbf{g}+\mathbf{S}_{d,m}. \](7.7) Here \(\mathbf{F}_{dg}\) is the force on the dispersed phase from the carrier. The additional \(p_{fr}\) term is not a thermodynamic pressure; it is a packing-activated regularization of the pressureless particulate subsystem.

The CMPS particle reconstruction uses \(T_d\) as the primitive thermal variable. A compact representation of the transported thermal balance is \[\frac{\partial(\rho_de_d)}{\partial t} +\nabla\cdot(\rho_de_d\mathbf{V}_d) =Q_{gd}+S_{d,E}, \qquad e_d=e_d(T_d). \](7.8) For a constant particle heat capacity over a limited temperature range, \(e_d\simeq c_{p,d}T_d\) up to the selected reference state. The source/Jacobian path uses the material properties and active thermal unknown rather than treating particle temperature as a passive 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 audited drag-work partition is identified as an open implementation issue in Chapter 20.

7.3 Drag coupling and particle response time

Define the slip velocity \[\mathbf{u}_r=\mathbf{u}-\mathbf{V}_d, \qquad U_r=|\mathbf{u}_r|.\] For spherical particles, the force per mixture volume associated with a conventional drag coefficient is \[\mathbf{F}_{dg} =\frac12 C_D\rho_g A_{\mathrm{proj}}U_r\mathbf{u}_r =\frac18 C_D\rho_g a_i U_r\mathbf{u}_r. \](7.9) The particle Reynolds number is \[\mathrm{Re}_d=\frac{\rho_gd_dU_r}{\mu_g}. \](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, \[C_D=\frac{24}{\mathrm{Re}_d}\left(1+0.15\,\mathrm{Re}_d^{0.687}\right),\] subject to the Reynolds-range and high-\(\mathrm{Re}\) branch used by the selected CMPS model option.

A useful relaxation scale follows by comparing the drag force with dispersed momentum. In the Stokes limit, \[\tau_v\sim\frac{\rho_{p,m}d_d^2}{18\mu_g}. \](7.11) Thus breakup can sharply decrease the local momentum-response time through its \(d_d^2\) dependence, while coalescence has the opposite effect.

7.4 Interphase heat transfer

The convective carrier-to-particle heat-transfer source is represented by \[Q_{gd}=h_da_i(T_g-T_d), \qquad h_d=\frac{k_g\mathrm{Nu}_d}{d_d}. \](7.12) For isolated spheres, a Ranz–Marshall form is \[\mathrm{Nu}_d=2+0.6\,\mathrm{Re}_d^{1/2}\mathrm{Pr}^{1/3} \](7.13) (Ranz and Marshall 1952). The corresponding thermal-response scale is approximately \[\tau_T\sim\frac{\rho_{p,m}c_{p,d}d_d^2}{6k_g\mathrm{Nu}_d}. \](7.14) Hence the IATE feedback is twofold: \(a_i\) 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 \(p_{fr}=0\) and sources are omitted, the one-dimensional dispersed mass–momentum subsystem is \[\frac{\partial}{\partial t} \begin{bmatrix}\rho_d\\ \rho_du\end{bmatrix} +\frac{\partial}{\partial x} \begin{bmatrix}\rho_du\\ \rho_du^2\end{bmatrix}=0.\] Both characteristic speeds collapse to \(u\), so the system is only weakly hyperbolic. CMPS regularizes this degeneracy only when the local concentration enters a near-packing regime.

Let \(\alpha_{fr}<\alpha_{\max}\) be the friction-pressure activation threshold. The regularization is inactive in the dilute region, \[p_{fr}=0, \qquad \alpha<\alpha_{fr}, \](7.15) and above the threshold uses a power-law form \[p_{fr}=\varepsilon_{\mathrm{eff}}\rho_d^{\gamma_p}. \](7.16) The effective coefficient is velocity-gated, \[\varepsilon_{\mathrm{eff}} =\min\!\left( \varepsilon_0, \frac{|\mathbf{V}_d|^2} {\gamma_p\rho_d^{\gamma_p-1}M_{\min}^2} \right), \](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 \[c_d^2=\frac{\partial p_{fr}}{\partial\rho_d} =\varepsilon_{\mathrm{eff}}\gamma_p\rho_d^{\gamma_p-1}, \](7.18) with the velocity-gated bound \[c_d\le\frac{|\mathbf{V}_d|}{M_{\min}}. \](7.19) The face-normal characteristic speeds then become \[\lambda_{1,2}=u_{n,d}\pm c_d, \](7.20) so strict hyperbolicity is restored wherever \(c_d>0\). The friction pressure supplies a finite signal speed; the separate receiver-side packing limiter in Chapter 8 enforces the hard bound \(\alpha\le\alpha_{\max}\).

7.6 One-group interfacial-area transport equation

The CMPS one-group IATE is \[\frac{\partial a_i}{\partial t} +\nabla\cdot(a_i\mathbf{V}_d) =S_{RC}+S_{BR}, \](7.21) with \[S_{BR}=S_{RT}+S_{TI}+S_G. \](7.22) Here \(S_{RC}\) is the random-collision or turbulence-induced coalescence contribution, \(S_{RT}\) is acceleration- or Rayleigh–Taylor-type breakup, \(S_{TI}\) is turbulent-impact breakup, and \(S_G\) 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 CMPS model options.

Since \([a_i]=\mathrm{m}^{-1}\), all IATE source terms have dimensions \[=[S_{RT}]=[S_{TI}]=[S_G] =\mathrm{m}^{-1}\mathrm{s}^{-1}. \](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 models use the carrier turbulence state when GE \(k\)–\(\omega\) RANS is active. In the CMPS implementation the local dissipation-rate proxy is formed as \[\epsilon_t=\beta^{*}k\omega, \](7.24) where the implementation coefficient used in the previously reviewed breakup path is \(\beta^{*}=0.09\). Dimensional consistency gives \[=\frac{\mathrm{m}^2}{\mathrm{s}^3}.\] The Kolmogorov scaling \(\epsilon_t^{1/3}d^{2/3}\) therefore provides a turbulent velocity scale at particle size \(d\).

The reviewed source paths activate the Luo-type coalescence and turbulence-driven breakup only when the dispersed model, IATE, and the required GE-RANS turbulence variables are enabled. If those prerequisites are not present, the corresponding source contribution is zero rather than being evaluated from undefined turbulence quantities.

7.8 Turbulence-induced coalescence

Coalescence merges dispersed entities and therefore reduces total area, \[S_{RC}<0.\] The CMPS 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 \[S_{RC}=-\Gamma_C\,\mathcal{C}(\alpha,a_i,d_d,\epsilon_t)\,\lambda_C, \](7.25) where \(\mathcal{C}\) is the collision-frequency/area-loss scale, \(\Gamma_C\) collects model constants, and \(0\le\lambda_C\le1\) 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 \(a_i\) back to \(d_d\), a negative \(S_{RC}\) increases the representative diameter at approximately fixed dispersed mass.

The active CMPS Luo1993 coalescence path is used only for GE-RANS + dispersed phase + IATE with the Luo1993 option selected. The source is assembled into the IATE residual and differentiated through the same AD path as the other local sources. This is important because the source depends on quantities that are themselves functions of \(k\), \(\omega\), \(\rho_d\), \(a_i\), carrier density, particle material density, and surface tension.

7.9 Turbulent-impact breakup

The previously reviewed CMPS turbulent-impact breakup path uses the turbulence dissipation proxy of Eq. (7.24). A breakup efficiency is written in the exponential form \[\lambda_B =\exp\!\left[ -\frac{K_B\sigma} {\rho_g d_d^{5/3}\epsilon_t^{2/3}} \right], \](7.26) where \(K_B\) is a model coefficient and \(\sigma\) is surface tension. The exponent is dimensionless because \[=\frac{\mathrm{kg}}{\mathrm{s}^2} =[\sigma].\] Thus stronger turbulence increases \(\lambda_B\), while larger surface tension suppresses breakup.

The implemented turbulent-impact area source has the dimensional structure \[S_{TI} =\Gamma_B\,\lambda_B\, \left(\frac{\alpha}{a_i}\right)^2 \alpha(1-\alpha)\, \epsilon_t^{1/3}d_d^{-11/3}\, \Phi_{\mathrm{pack}}, \](7.27) where \(\Gamma_B\) denotes the model coefficient used by the selected closure and \(\Phi_{\mathrm{pack}}\) denotes the safeguarded packing-related factor in the implementation. The source path uses a protected remaining-packing measure based on \[\max\!\left(|\alpha_{\max}-\alpha|,\epsilon_{num}\right) \](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 implementation detail rather than silently replacing it by an unsafeguarded analytical expression.

Equation (7.27) has the required units: \((\alpha/a_i)^2\) contributes \(\mathrm{m}^2\), \(\epsilon_t^{1/3}\) contributes \(\mathrm{m}^{2/3}\mathrm{s}^{-1}\), and \(d_d^{-11/3}\) contributes \(\mathrm{m}^{-11/3}\), giving \(\mathrm{m}^{-1}\mathrm{s}^{-1}\) before dimensionless coefficients and packing factors.

The turbulent-impact source is positive when active, \[S_{TI}>0,\] and therefore increases \(a_i\) and reduces \(d_d\).

7.10 Acceleration/Rayleigh–Taylor-type breakup

Strong relative acceleration can destabilize a dense particle or droplet against the surrounding gas. CMPS 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 \[d_{RT,crit}\propto \sqrt{\frac{\sigma}{|\rho_{p,m}-\rho_g|\,a_{rel}}}, \](7.29) where \(a_{rel}\) is the relevant relative/body acceleration magnitude used by the closure.

The breakup source can be represented structurally as \[S_{RT}=\Gamma_{RT}\,\mathcal{R}_{RT} \left(a_i,d_d,d_{RT,crit},\ldots\right)\, \mathcal{A}_{RT}, \](7.30) where \(\mathcal{A}_{RT}\) is a smooth activation function. The current formulation uses a smooth transition near the critical condition rather than a hard discontinuous source switch; a \(\tanh\)-type activation is appropriate for preserving a usable AD derivative. The source is positive when \(d_d\) exceeds the stable size selected by the model.

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

7.11 Gravity/Eotvos-type breakup

For body-force-dominated deformation, CMPS includes an Eotvos-type breakup branch. The controlling dimensionless group is \[\mathrm{Eo} =\frac{|\rho_g-\rho_{p,m}|\,g_{eff}\,d_d^2}{\sigma}, \](7.31) where \(g_{eff}\) is the acceleration magnitude used by the model. Rearranging a critical value \(\mathrm{Eo}_{crit}\) gives the corresponding stable-diameter scale \[d_{G,crit} =\sqrt{\frac{\mathrm{Eo}_{crit}\sigma} {|\rho_g-\rho_{p,m}|g_{eff}}}. \](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 \[S_G =\frac{a_{i,crit}-a_i}{\tau_G}\,\mathcal{A}_{G}, \qquad a_{i,crit}=\frac{6\alpha}{d_{G,crit}}, \](7.33) with \(\mathcal{A}_{G}\) denoting the model activation. The sign is positive in the breakup regime because \(a_{i,crit}>a_i\) 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, \[S_{IATE}=S_{RC}+S_{RT}+S_{TI}+S_G. \](7.34) A useful local size-evolution identity follows from Eq. (7.4). Neglecting material-density variation for clarity, \[\frac{1}{d_d}\frac{D d_d}{Dt} =\frac{1}{\rho_d}\frac{D\rho_d}{Dt} -\frac{1}{a_i}\frac{D a_i}{Dt}. \](7.35) For a homogeneous parcel with no net dispersed-mass source, an IATE source therefore gives approximately \[\left.\frac{1}{d_d}\frac{D d_d}{Dt}\right|_{IATE} \simeq-\frac{S_{IATE}}{a_i}. \](7.36) This makes the physical interpretation immediate: breakup (\(S_{IATE}>0\)) decreases diameter, while coalescence (\(S_{IATE}<0\)) 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,

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

• divisions by \(a_i\), \(d_d\), or remaining packing margin must use the implementation 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 Implicit coupling and AD

Particle convection, friction pressure, drag, heat transfer, gravity, IATE transport, and breakup/coalescence are assembled in the same coupled residual framework as the carrier equations. Forward-mode AD differentiates the active expressions with respect to carrier and dispersed primitive variables. Consequently, the IATE equation is not only coupled to \(a_i\): its Jacobian can contain derivatives with respect to \(\rho_d\), \(T_d\), carrier density/temperature, \(k\), \(\omega\), and any other active quantity used by the selected closure.

For a local IATE source \(S_{IATE}(\mathbf{q}_g,\mathbf{q}_d)\), \[\delta S_{IATE} =\frac{\partial S_{IATE}}{\partial\mathbf{q}_g}\delta\mathbf{q}_g +\frac{\partial S_{IATE}}{\partial\mathbf{q}_d}\delta\mathbf{q}_d. \](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

The current breakup-model selector exposes a Reitz1999 branch. In the present source, that selector enters an acceleration-driven RT-style interfacial-area source rather than a classical standalone wavelength-only KH correlation. The implementation computes slip-driven aerodynamic acceleration from the spherical projected-area density \(A_p=a_i/4\),

It then forms

and smoothly activates breakup when \(d_p\) exceeds \(\lambda_{RT}\). 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.