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TrioCFD 1.9.8
TrioCFD documentation
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TrioCFD 1.9.8
TrioCFD documentation
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A fundamental aspect of modelling the interaction between the liquid and gas phases is precisely predicting the concentration of interfaces. Some models rely on the interfacial area concentration ( \(a_i\)) or an equivalent diameter ( \(D_{sm}\)). The dispersed fluid is depicted as a collection of bubbles with diverse diameters, distinguished by a distribution of bubble diameters and the concentration of interfaces. The Sauter mean diameter ( \(D_{sm}\)) of the distribution \(f_d\) of sizes \(D\) is:
\[D_{sm}=\frac{\int f_d D^3\, dD}{\int f_d D^2\, dD}=6\frac{\int f_d V\, dV}{\int f_d A_i\, dV}=6\frac{\alpha}{a_i} \]
using the area concentration per unit volume \(A_i=\pi D^2\). To represent the dispersed phase it is sufficient to impose an equivalent diameter, but it is also possible to add a transport equation for the interfacial area concentration.
Uniform diameter (Diametre_bulles_constant, param diametre): a constant diameter is imposed at every point in space, assigned to all phases except the liquid phase.
Non-uniform diameter (Diametre_bulles_champ): a user-defined, spatially varying diameter is imposed through the TRUST Champ_Don field framework.
Two separated size-group methods are popular for predicting interfacial area concentration: one based on an arbitrary number of groups to reproduce a distribution (MUSIG or i-MUSIG [180], [28], [108]) and the other reproducing the distribution through the Sauter mean diameter (IATE). The generalized Interfacial Area Transport Equation (IATE) was developed by Kocamustafaogullari and Ishii [92], [93]. For adiabatic flows, with \(\psi^{internal}_{j}\) a source term and \(\psi^{intergroup}_j\) an intergroup term:
\[\frac{\partial a_{i}}{\partial t} + \underbrace{\nabla\cdot(\mathbf{U} a_{i})}_{\text{convection}} = \underbrace{\frac{2}{3}\frac{a_{i}}{\alpha}\frac{D\alpha}{Dt}}_{\text{volume change}} + \underbrace{\sum_j \psi^{\text{intergroup}}_{j}}_{\text{intergroup}} + \underbrace{\sum_j \psi^{\text{internal}}_{j}}_{\text{intragroup}} \]
Using \(\frac{D\alpha\rho_g}{Dt}=\alpha\frac{d\rho_g}{dt}+\rho_g\frac{D\alpha}{Dt}=\Gamma_{transfer}+\Gamma_{nucleation}\), one substitutes \(\frac{D\alpha}{Dt}=\frac{1}{\rho_g}(\Gamma_{\text{transfer}}+\Gamma_{\text{nucleation}}-\alpha\frac{d\rho_g}{dt})\) to obtain:
\[\frac{\partial a_{i}}{\partial t} + \nabla\cdot(\mathbf{U} a_{i}) = \underbrace{-\frac{2}{3}\frac{a_{i}}{\rho_g}\frac{d\rho_g}{dt}}_{\text{density change}} + \underbrace{\frac{2}{3}\frac{a_{i}}{\alpha}\frac{\Gamma_{\text{transfer}}}{\rho_g}}_{\text{condensation}} + \underbrace{\frac{2}{3}\frac{a_{i}}{\alpha}\frac{\Gamma_{\text{nucleation}}}{\rho_g}}_{\text{nucleation}} + \sum_j \psi^{\text{intergroup}}_{j} + \sum_j \psi^{\text{internal}}_{j} \]
The density change model \(-\frac{2}{3}\frac{a_{i}}{\rho_g}\frac{d\rho_g}{dt}\) is implemented in Variation_rho_Elem_PolyMAC_P0, filling the \(M_{a_i}\), \(M_T\), \(M_P\) matrices and secmem with the discrete (chain-rule) derivatives with respect to temperature and pressure. The condensation model \(\frac{2}{3}\frac{a_i}{\alpha\rho_g}G\) (with \(G\) a correlation) is implemented in Source_Flux_interfacial_base.
The model is described in [188] :
\[\frac{\partial a_{i}}{\partial t} + \underbrace{\nabla\cdot(\mathbf{U} a_{i})}_{\text{convection}} = \underbrace{\frac{2}{3}\frac{a_{i}}{\alpha}\left(\frac{\Gamma_{\text{cond}}}{\rho_g}-\frac{\alpha_g}{\rho_g}\frac{d\rho_g}{dt}\right)}_{\text{volume change}} + \underbrace{\pi d_{\text{dep}}^2\Phi_N}_{\text{nucleation}} + \underbrace{\frac{36\pi}{3}\left(\frac{\alpha}{a_i}\right)^2\Phi_{\text{coal}}}_{\text{coalescence}} + \underbrace{\frac{36\pi}{3}\left(\frac{\alpha}{a_i}\right)^2\Phi_{\text{breakup}}}_{\text{break-up}} \]
Coalescence (Coalescence_bulles_1groupe_Yao_Morel):
\[\frac{36\pi}{3}\left(\frac{\alpha}{a_i}\right)^2\Phi_{\text{Coal}} = \frac{\pi}{3\times 6^{5/3}}\alpha^{1/3}a_i^{5/3}\varepsilon^{1/3}\cdot K_{c1}\frac{-1}{g(\alpha)+K_{c2}\alpha\sqrt{We/We_{cr}}}\exp\left(-K_{c3}\sqrt{\frac{We}{We_{cr}}}\right) \]
with \(K_{c1}=2.86\), \(K_{c2}=1.922\), \(K_{c3}=1.017\), \(We_{cr}=1.24\), \(g(\alpha)=\frac{\alpha_\text{max}^{1/3}-\alpha^{1/3}}{\alpha_\text{max}^{1/3}}\), \(\alpha_\text{max}=\frac{\pi}{6}\), \(\beta_k=0.09\) (default).
Break-up (Rupture_bulles_1groupe_Yao_Morel):
\[\frac{36\pi}{3}\left(\frac{\alpha}{a_i}\right)^2\Phi_{\text{breakup}} = \frac{\pi}{3\times 6^{5/3}}\alpha^{-2/3}(1-\alpha)a_i^{5/3}\varepsilon^{1/3}\cdot K_{b1}\frac{1}{1+K_{b2}(1-\alpha)\sqrt{We/We_{cr}}}\exp\left(-\frac{We}{We_{cr}}\right) \]
with \(K_{b1}=1.6\), \(K_{b2}=0.42\), \(We_{cr}=1.24\).
Nucleation (Nucleation_paroi_PolyMAC_P0, injected only on boundary elements, fully explicit):
\[\pi d_{dep}^2\Phi_N=\pi d_{dep}^2\frac{\Phi_{e}}{L_{\text{vap}}\rho_g\frac{\pi}{6}d_\text{dep}^3}=6\frac{\Phi_{\text{nucleation}}}{\max(d_{\text{nuc}},10^{-8})\rho_g L_{\text{vap}}} \]
with \(\Phi_{e}\) the wall heat transfer.
A particular case considers two groups of bubbles, separated by the critical diameter \(D_{smc}=4\sqrt{\frac{\sigma}{g(\rho_l-\rho_g)}}\) (e.g. the experimental limit between quasi-spherical and distorted bubbles). For each group \(g1\), \(g2\), the transport equation has the same structure as the 1-group equation, augmented with density-sliding and mass-transfer-sliding terms that exchange bubbles between groups, plus intergroup/intragroup sources. The inter-group mass transfers \(\Gamma_{g1}\), \(\Gamma_{g2}\) combine an intergroup term, a density-group-shift term, a condensation term and a nucleation term, weighted by \(\chi_d(D_{smc}/D_{sm1})^3\) ( \(\chi_d=1\) for a uniform distribution profile). During the averaging process [87], a diffusion term \(K\sqrt{u'^2}D_{sm}\nabla a_i=K\sqrt{\frac{2k}{3}}D_{sm}\nabla a_i\) (with \(K=1/3\)) and a lift term emerge; these are not yet fully validated [142].
All source-term models are based on five mechanism categories: Random Collisions (RC), Wake Entrainment (WE), Turbulent Impacts (TI), Shearing-off (SO) and Surface Instability (SI). RC is a coalescence phenomenon where two bubbles collide and merge because of a turbulent eddy of comparable size; WE happens when a smaller bubble in the wake of a bigger one accelerates and collides with it; TI is break-up due to turbulent eddies; SO is break-up from the shearing-off of cap bubbles; SI is break-up of large bubbles due to surface instability.
The number of processes and the dimensionless coefficients differ between models:
| Coefficient | Sun | Smith | Schlegel | Fu | Dave |
|---|---|---|---|---|---|
| \(C^{(1)}_{RC}\) | 0.005 | 0.01 | 0.01 | 0.0041 | 0.26 |
| \(C^{(12,2)}_{RC}\) | 0.005 | 0.01 | 0.05 | 0.005 | 0.41 |
| \(C^{(2)}_{RC}\) | 0.005 | 0.01 | 0.01 | 0.005 | 1.00 |
| \(C_{RC0}\) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
| \(C_{RC1}\) | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |
| \(\alpha_{g1,max}\) | 0.62 | 0.62 | 0.62 | 0.75 | 0.62 |
| \(C^{(1)}_{WE}\) | 0.002 | 0.002 | 0.002 | 0.002 | 0.001 |
| \(C^{(12,2)}_{WE}\) | 0.002 | 0.01 | 0.02 | 0.015 | 0.017 |
| \(C^{(2)}_{WE}\) | 0.005 | 0.01 | 0.05 | 10. | 0.021 |
| \(C^{(1)}_{TI}\) | 0.1 | 0.05 | 0.05 | 0.0085 | 0.013 |
| \(C^{(12,2)}_{TI}\) | 0.02 | 0.04 | 0.02 | 0.02 | 0.006 |
| \(C^{(2)}_{TI}\) | 0.02 | 0.01 | 0.01 | 0.02 | 0.023 |
| \(We_{cr1}\) | 6.5 | 1.2 | 1.2 | 6.0 | 6.0 |
| \(We_{cr2}\) | 7.0 | 1.2 | 1.2 | 6.0 | 6.0 |
| \(C_{SO}\) | \(3.8\times10^{-5}\) | \(2.5\times10^{-5}\) | \(5.0\times10^{-5}\) | 0.031 | \(1.4\times10^{-5}\) |
| \(We_{c,SO}\) | 4500 | 4000 | 10 | 4500 | 4500 |
The detailed RC, WE, TI, SO and SI source/sink term expressions \(\phi^{(\cdot)}\) and \(\eta^{(\cdot)}\) (Smith model) follow the formulations of [162]. As an example, the Random-Collision Group-1 sink and the leading Wake-Entrainment and Turbulent-Impact terms are:
\[\phi_{RC}^{(1)} = -0.17\, C_{RC}^{(1)}\lambda_{RC}^{(1)}\frac{\varepsilon^{1/3}\alpha_{g1}a_{i1}^{5/3}}{\alpha_{g1,max}^{1/3}(\alpha_{g1,max}^{1/3}-\alpha_{g1}^{1/3})}\left[1-\exp\left(-C_{RC1}\frac{\alpha_{g1,max}^{1/3}\alpha_{g1}^{1/3}}{\alpha_{g1,max}^{1/3}-\alpha_{g1}^{1/3}}\right)\right] \]
\[\phi_{WE}^{(1)} = -0.17\, C_{WE}^{(1)}C_{D1}^{1/3}U_{r1}a_{i1}^2, \qquad \phi_{TI}^{(1)} = 0.12\, C_{TI}^{(1)}\varepsilon^{1/3}(1-\alpha_g)\frac{a_{i1}^{5/3}}{\alpha_{g1}^{2/3}}\exp\left(-\frac{We_{cr1}}{We_1}\right)\sqrt{1-\frac{We_{cr1}}{We_1}} \]
with \(\lambda_{RC}^{(1)}=\exp(-C_{RC0}\frac{D_{sm1}^{5/6}\rho_l^{1/2}\varepsilon^{1/3}}{\sigma^{1/2}})\), \(We_1=\frac{2\rho_l\varepsilon^{2/3}D_{sm1}^{5/3}}{\sigma}\), and the drag-related quantities \(C_{D1}=\frac{2}{3}D_{sm1}\sqrt{\frac{g\Delta\rho}{\sigma}}\left(\frac{1+17.67[f(\alpha_{g1})]^{6/7}}{18.67 f(\alpha_{g1})}\right)^2\), \(f(\alpha_{g1})=(1-\alpha_{g1})^{1.5}\).
1.16.1