Homogeneous Mixture model

Homogeneous Mixture model#

This chapter focuses on the modeling of homogeneous mixtures through TrioCFD multiphase. The first part is a review of the use of the evanescence operator ; the second part describes the equilibrium models dedicated to mixtures.

Homogeneous evanescence for mixture modeling#

The Pb_multiphase framework allows to model a mixture by taking advantage of the evanescence operator described in the [dedicated section] REF sec:evanescence. By choosing the correct values of alpha_res to 1 and alpha_res_min to 0.5, one can reduce the system to three partial differential equations corresponding to the dynamics of a mixture. Without any model, this approach can be referred to as the Homogeneous Equilibrium Model as the two phases can be considered dynamically locked and at the same temperature. This can be the starting point of a homogeneous relaxation model approach by adding different models to describe the relation between the two phases.

Imposing the above-mentioned values lead to a simplification of Equation REF eq:evanescence_momentum:

(248)#\[\begin{equation} \newcommand{\parent}[1]{\left(#1\right)} \definecolor{myteal}{RGB}{0,128,128} \definecolor{myslateblue}{RGB}{106,90,205} \definecolor{mydarkorchid}{RGB}{153,50,204} \label{eq:evanescence_momentum_HEM} \begin{pmatrix} {\mathcal{Q}_{\text{pred}}} \\ {\mathcal{Q}_{\text{mino}}} \end{pmatrix} \Longrightarrow \begin{pmatrix} {\mathcal{Q}_{\text{pred}}} + {\mathcal{Q}_{\text{mino}}} \\ {v_{\text{mino}} = v_{\text{pred}} + v_{\text{drift}}} \end{pmatrix} \end{equation}\]

Dedicated mixture modeling#

Drift velocity#

The drift velocity of a gas phase \(u_{gj}\) is defined as the velocity of the gas phase \(u_g\) with respect to the volume center of the mixture \(j\). The area average \(\prec F\succ \) of a quantity \(F\) over the cross-sectional area \(A\) is defined by:

(249)#\[\begin{equation} \prec F\succ=\frac{1}{A}\int_A FdA \end{equation}\]

The one-dimensional drift-flux model is defined as:

(250)#\[\begin{equation} \frac{\prec j_g\succ}{\prec \alpha_g \succ}=C_0 \prec j\succ+V_{g0}, \end{equation}\]

with

(251)#\[\begin{align} &C_0 = \frac{\prec \alpha_g j \succ}{\prec \alpha_g \succ \prec j \succ},\\ &V_{g0} = \frac{\prec \alpha_g u_{gj} \succ}{\prec \alpha_g \succ}. \end{align}\]

The model is implemented in:

void Vitesse_derive_base::set_param(Param& param)

The drift velocity operator must fill \(vr\) and \(dvr\) tabs for each dimension \(d\) so that:

\(\texttt{vr}({\color{myteal}k1}, {\color{mydarkorchid}k2}, d)\) Relative velocity in dimension d
\(\texttt{dvr}({\color{myteal}k1}, {\color{mydarkorchid}k2}, d, \texttt{dimension}*{\color{mydarkorchid}k2}+d)\) Relative velocity derivative regarding the inlet superficial velocity \(\texttt{v}(d, {\color{mydarkorchid}k2})\)
\(\texttt{vr}({\color{mydarkorchid}k2}), {\color{myteal}k1}, d)=-\texttt{output.vr}({\color{myteal}k1}, {\color{mydarkorchid}k2}), d)\)
\(\texttt{dvr}({\color{myteal}k1}, {\color{mydarkorchid}k2}), d, \texttt{dimension}*{\color{mydarkorchid}k2}+d)=-\texttt{vr}({\color{myteal}k1}, {\color{mydarkorchid}k2}), d, \texttt{dimension}*{\color{mydarkorchid}k2}+d)\)

The following table summarises the availability of drift velocity models in TRUST/TrioCFD:

Model	Used	Validated	Test case
Constant bubble	Yes	Yes	TRUST : canal bouillant drift & TrioCFD : Drift flux
Ishii-Hibiki	Yes	Yes	TRUST : canal bouillant drift & TrioCFD : Drift flux
Spelt	Yes	No
Forces	Yes	Yes	TRUST : canal bouillant drift & TrioCFD : Drift flux

Constant#

The model is described in Ishii [Ish77] and is implemented in:

void Vitesse_derive_constante::set_param(Param& param)
{
  param.ajouter("C0", &C0, Param::REQUIRED);
  param.ajouter("vg0_x", &vg0[0], Param::REQUIRED);
  param.ajouter("vg0_y", &vg0[1], Param::REQUIRED);
  if (dimension == 3) param.ajouter("vg0_z", &vg0[2], Param::REQUIRED);
}

Ishii-Hibiki : Bubbly flow#

The model is described in Hibiki and Ishii [HI02] and is implemented in:

void Vitesse_derive_Ishii::set_param(Param& param)
{
  param.ajouter("subcooled_boiling", &sb_, Param::REQUIRED);
}

Default values:

\(\texttt{sb_} = 0\) (0: no, 1: yes),
\(\texttt{Cinf} = 1.2\),
\(\texttt{theta} = 1.75\),
\(\texttt{zeta} = 18.0\).

The implemented model is:

(252)#\[\begin{align} & C_0=\parent{\texttt{Cinf}+\parent{1-\texttt{Cinf}} \sqrt{\frac{\rho_g}{\rho_l}}}\parent{1-\texttt{sb_}\ exp(-\texttt{zeta}\times\alpha_g)}\\ & \vec{V_{g0}} =-\sqrt{2}\parent{\frac{\parent{\rho_l-\rho_g}g\sigma}{\rho_l^2}}^{1/4}\parent{1-\alpha_g}^{\texttt{theta}} \frac{\vec{g}}{|g|} \end{align}\]

Spelt Biesheuvel#

The model is described in Spelt and Biesheuvel [SB97] and is implemented in:

void Vitesse_derive_Spelt_Biesheuvel::set_param(Param& param)

Default values \(\texttt{Prt_} = 1.\)

(253)#\[\begin{equation} \vec{V_{g0}} =\parent{-(u_g-u_l)\frac{\vec{g}}{|g|}+\frac{\nu_{\text{Spelt}}\nabla \alpha_g}{\max(\alpha_g,0.0001)}}\parent{1-C_0\alpha_g} \end{equation}\]

with

\(C_0=1.0\),
\(\nu_{\text{Spelt} = \frac{\parent{C_\mu^{1/4}k^{1/2}}^2L }{u_g-u_l}\parent{\frac{1}{2}+\frac{3}{8}\parent{\frac{\tau^2}{\lambda_t}}^2\frac{\lambda_t}{L}}\),
\(L=C_\mu^{3/4}\frac{k^{3/2}}{\varepsilon}\),
\(\tau=\frac{u_g - u_l}{2g}\),
\(\lambda_t=\sqrt{10\nu_l\frac{k}{\varepsilon}}\),
\(\varepsilon=C_\mu \frac{k^2}{\nu_t}\).

Forces#

The model is implemented in:

void Vitesse_derive_Forces::set_param(Param& param)
{
  param.ajouter("alpha_lim", &alpha_lim_);
}

Default values : \(\texttt{alpha_lim_}=10^{-5}\).

(254)#\[\begin{equation} \vec{V_{g0}} =\parent{-\parent{u_g-u_l}\frac{\vec{g}}{|g|}+\frac{F^{\text{dispersion}}+F^{\text{lift}}}{f^DU_r}}\parent{1-C_0\alpha_g} \end{equation}\]

Two-phase frictional multiplier#

The frictional pressure drop in gas–liquid flow can be expressed as a function of a two-phase friction multiplier [FSCH22], based on empirical correlations and both pure liquid friction \(f_l\) and pure gas friction \(f_g\).

The general expression of the two-phase frictional multiplier is:

(255)#\[\begin{equation} \Phi_{k}^2=\frac{(\frac{\partial p}{\partial z})|_{\text{Two-phase}}}{(\frac{\partial p}{\partial z})|_{\text{Single phase k}}} \end{equation}\]

void Multiplicateur_diphasique_base::set_param(Param& param)

The available input parameters are:

const double alpha ; // Void fraction
const double rho ; //  Density
const double v ;  //Velocity
const double f ;  // Darcy coefficient as if all the flow rate was in phase k
const double mu ;  // Viscosity
const double Dh ;  // Hydraulic diameter
const double gamma ;  // Surface tension

The interfacial heat flux operator must fill \(\mathit{coeff}\) tab so that:

\(\mathit{coeff}(k, 0)\) multiplier for the single phase friction factor
\(\mathit{coeff}(k, 1)\) multiplier for the mix friction factor

Homogeneous#

The model is implemented in :

void Multiplicateur_diphasique_homogene::set_param(Param& param)
{
  param.ajouter("alpha_min", &alpha_min_);
  param.ajouter("alpha_max", &alpha_max_);
}

Default values : \(\texttt{alpha_min_}= 0.9995\), \(\texttt{alpha_max_} = 1\).\ The model implemented is :

(256)#\[\begin{align} & \Phi^2= 1+x\parent{\frac{\rho_l}{\rho_g}-1}\\ & \mathit{coeff}(n_l, 0) = \text{Frag}_l\times\Phi^2,\\ & \mathit{coeff}(n_g, 0) = \frac{\text{Frag}_g}{\alpha_g^2} \end{align}\]

with

\(\text{Frag}_g = \min\parent{\max\parent{\frac{\alpha_g-\texttt{alpha_min_}}{\texttt{alpha_max_}-\texttt{alpha_min_}},0},1}\)
\(\text{Frag}_l=1-\text{Frag}_g\)

Fridel: horizontal and vertical smooth tubes with \(\mu_l/\mu_g<1000\)#

The model is described in Friedel [Fri79] and is implemented in:

void Multiplicateur_diphasique_Friedel::set_param(Param& param)
{
  param.ajouter("alpha_min", &alpha_min_);
  param.ajouter("alpha_max", &alpha_max_);
  param.ajouter("min_lottes_flinn", &min_lottes_flinn_);
  param.ajouter("min_sensas", &min_sensas_);
}

Default values:

\(\texttt{alpha_min_} = 1\),
\(\texttt{alpha_max_} = 1.1\),
\(\texttt{min_lottes_flinn_} = 0\),
\(\texttt{min_sensas_} = 0\).

The model implemented is:

(257)#\[\begin{equation} \Phi^2= E+\frac{3.24FH}{Fr^{0.0454}We^{0.035}} \end{equation}\]

if \(F_k \min\parent{1, 1.14429\alpha_l^{0.6492}} < \Phi^2 F_m \alpha_l^2\) and \(\texttt{min_sensas}=1\) and \(\texttt{min_lottes_flinn_}=1\):

(258)#\[\begin{align} &\mathit{coeff}(n_l, 0) = \frac{\text{Frac}_l}{\alpha_l^2},\\ &\mathit{coeff}(n_g, 0) = \frac{Frac_g}{\alpha_l^2}, \end{align}\]

else

(259)#\[\begin{align} &\text{coeff}(n_l, 1) = \text{Frag}_l\times\Phi^2,\\ &\text{coeff}(n_g, 1) = \text{Frag}_g\times\Phi^2 \end{align}\]

with

\(\text{Frag}_g=\min\parent{\max\parent{\frac{\alpha_g - \texttt{alpha_min_}}{\texttt{alpha_max_} - \texttt{alpha_min_}}, 0}, 1}\),
\(\text{Frag}_l=1-\text{Frag}_g\),
\(E=(1-x)^2+x^2\frac{\rho_lf_g}{\rho_gf_l}\)
\(F=x^{0.78}(1-x)^{0.224}\)
\(G=\alpha_l\rho_lu_l+\alpha_g\rho_gu_g\)
\(H=(\frac{\rho_l}{\rho_g})^{0.91}(\frac{\mu_g}{\mu_l})^{0.19}(1-\frac{\mu_g}{\mu_l})^{0.7}\)
\(x=\frac{\alpha_g \rho_gu_g}{G}\)
\(\mathit{Fr}=\frac{\parent{\alpha_l\rho_lu_l+\alpha_g \rho_gu_g}^2}{9.81D_h\rho_m^2}\)
\(\mathit{We}=\frac{G^2D_h}{\sigma \rho_m}\)
\(\rho_m=\frac{1}{\frac{x}{\rho_g}+\frac{1-x}{\rho_l}}\)

Lottes and Flinn: sodium two-phase pressure drop#

The model is described in Lottes and Flinn [LF56] and is implemented in:

void Multiplicateur_diphasique_Lottes_Flinn::set_param(Param& param)
{
  param.ajouter("alpha_min", &alpha_min_);
  param.ajouter("alpha_max", &alpha_max_);
}

Default values : \(\texttt{alpha_min_} = 0.9\), \(\texttt{alpha_max_} = 0.95\). \ The model implemented is :

(260)#\[\begin{equation} \mathit{coeff}(n_l, 0) = \begin{cases}\frac{\max\parent{\alpha_l - 1+ \texttt{alpha_max_}, 0}}{\parent{1-\texttt{alpha_min_}}^2},\text{ if }\alpha_l<1-\texttt{alpha_min_}\\ \frac{1}{\alpha_l^2},\text{ otherwise}. \end{cases}, \end{equation}\]

(261)#\[\begin{equation} \mathit{coeff}(n_g, 0) = \min\parent{\max\parent{\frac{\alpha_g-\texttt{alpha_min_}}{\texttt{alpha_max_}-\texttt{alpha_min_}}, 0} ,1} \end{equation}\]

Muller-Steinhagen: air–water, water-hydrocarbons and refrigerants in pipes#

The model is described in Müller-Steinhagen and Heck [MullerSH86] and is implemented in:

void Multiplicateur_diphasique_Muhler_Steinhagen::set_param(Param& param)
{
  param.ajouter("alpha_min", &alpha_min_);
  param.ajouter("alpha_max", &alpha_max_);
  param.ajouter("min_lottes_flinn", &min_lottes_flinn_);
  param.ajouter("min_sensas", &min_sensas_);
  param.ajouter("a", &a_);
  param.ajouter("b", &b_);
  param.ajouter("c", &c_);
}

Default values:

\(\texttt{alpha_min_} = 1\),
\(\texttt{alpha_max_} = 1.1\)
\(\texttt{a_} = 2\)
\(\texttt{b_} = 1\)
\(\texttt{c_} = 3\)
\(\texttt{min_lottes_flinn_} = 0\)
\(\texttt{min_sensas_} = 0\).

The model implemented is:

if
- \(F_k min(1,1.14429\alpha_l^{0.6492}) < \rho_l F_m \alpha_l^2 \frac{f_m^*}{f_l}\)
- \(\texttt{min_sensas}=1\)
- \(\texttt{min_lottes_flinn_}=1\)
- then:

(262)#\[\begin{equation} \mathit{coeff}(n_l, 0) = \frac{\min\parent{1, 1.14429\alpha_l^{0.6492}}}{\alpha_l^2}. \end{equation}\]

else

(263)#\[\begin{align} & \mathit{coeff}(n_l, 1) = \text{Frac}_l\times\rho_l\frac{f_m^*}{f_l},\\ & \mathit{coeff}(n_g, 1) = \text{Frac}_g\rho_g\frac{f_m^*}{f_g} \end{align}\]

with

\(\text{Frac}_g=\min\parent{\max\parent{\frac{\alpha_g-\texttt{alpha_min_}}{\texttt{alpha_max_}-\texttt{alpha_min_}},0},1}\)
\(\text{Frac}_l=1-\text{Frac}_g\),
\(G=\alpha_l\rho_lu_l+\alpha_g\rho_gu_g\)
\(x=\frac{\alpha_g \rho_gu_g}{G}\)
\(fm^*=\parent{\frac{f_l}{\rho_l}+ax^b\parent{\frac{f_g}{\rho_g}-\frac{f_l}{\rho_l}}}(1-x)^{1/c}+\frac{f_g}{\rho_g}x^c.\)

Homogeneous Mixture model

Contents

Homogeneous Mixture model#

Homogeneous evanescence for mixture modeling#

Dedicated mixture modeling#

Drift velocity#

Constant#

Ishii-Hibiki : Bubbly flow#

Spelt Biesheuvel#

Forces#

Two-phase frictional multiplier#

Homogeneous#

Fridel: horizontal and vertical smooth tubes with \(\mu_l/\mu_g<1000\)#

Lottes and Flinn: sodium two-phase pressure drop#

Muller-Steinhagen: air–water, water-hydrocarbons and refrigerants in pipes#