Association

The implementation of association model in ℜeos is based on the work of Michelsen 2001 and Michelsen 2006. The dimensionless association contribution to the Helmholtz free energy is given by:

\[ a^{asc} = \sum_j^S m_j \left( \ln X_j - \frac{X_j}{2} + \frac{1}{2} \right) M_j \]

where $S$ is the number of distinguishable sites in the mixture, $m_j$ is the molar fraction of site $j$ and $M_j$ is the multiplicity of site $j$.

The fraction of non-bonded sites $X_j$ is obtained by solving implicitily:

$$ X_j = \frac{m_j}{m_j + \sum_l^S K_{jl} X_l M_l} $$ where $K_{jl}$ is the association constant of sites $j$ and $l$: $$ K_{jl} = \rho m_j m_l \Delta_{jl} $$ and $\Delta^{jl}$ is the association strength between sites $j$ and $l$. Because the expressions for $\Delta^{jl}$ depend on the specific association model, we can use the general expression of the dimensionless association strength: $$ \delta^{jl} = \kappa^{jl} \left[ \exp\left( \frac{\epsilon^{jl}}{RT} \right) - 1 \right] $$ where $\kappa^{jl}$ is the association volume and $\epsilon^{jl}$ is the association energy between sites $j$ and $l$. Then,

$$ \Delta^{jl} = f_v ^{ik} \delta^{jl} $$ where $f_v^{ik}$ is a volumectric factor that depends on the specific model and the the combining rule choosen for the components hosts $i$ and $k$ of sites $j$ and $l$, respectively.

Model	Volumetric factor $f_v^{ik}$
PC-SAFT	$d_{ik}^3\, g_{ik}^{seg}$
CPA	$b_{ik}\, g(\rho)$

Association sites

Each site $s_j$ has a owner $i$, a type $\alpha$, an index $j$, a multiplicity $M_j$ and the parameters $\epsilon^{j}$ and $\kappa^{j}$. The type of each site defines the allowed interactions with other sites.

Then, to satisfy all association schemes proposed by Huang and Radosz 1990, we propose 3 types of sites: $A, B$ and $C$. $A$ and $B$ sites can interact with all types except with sites of the same type, while $C$ sites can interact with all types, including themselves.

Example: Association parameters with pure water 4C in Reos

AssociativeParameters(
na=1, nb=1, nc=0,
sites=[
   Site(type=A,owner=0,idx=0,mul=2,eps=16655,kappa=0.0692),
   Site(type=B,owner=0,idx=1,mul=2,eps=16655,kappa=0.0692)],
interactions=[
   SiteInteraction(j=0,l=1,epsilon=16655,kappa=0.0692, rule='cr1')
   ])

Calculation of unbonded site fractions

To calculate the fractions of unbonded sites we can use the successive substitution method or the second-order method proposed by Michelsen 2006:

\[ g_j = \frac{\partial Q}{\partial X_j} = m_j \left( \frac{1}{X_j} - 1 \right) - \sum_{l=1}^{S} K_{jl} X_l M_l \]

\[ H_{jl} = -\delta_{jl}\left( \frac{m_{j}}{X_{j}^2} \right) - K_{jl} \]

Otherwise, in simple cases, the fractions of unbonded sites can be calculated analytically.

If the mixture contains only 1 site of type $A$ and 1 site of type $B$, the fractions of unbonded sites can be calculated analytically:

\[ X_2 = \frac{2 m_1 m_2 }{m_{1} m_2 + k\left(m_1 M_1 - m_2 M_2\right) + \sqrt{4km_2m_1^2M_1 + \left[m_1m_2 + k \left( m_1 m_2 M_1 M_2 \right) \right]^2}} \]

$$ X_1 = \frac{m_1}{m_1 + k X_2 M_2} $$ where $k = K_{12} = K_{21}$.

If the mixture contains only 1 site of type $C$, the fraction of unbonded fraction is given by:

\[ X_1 = \frac{2 m_1}{m_1 + \sqrt{4 k M_1 m_1 + m_1^2}} \]

where $k = K_{11}$.

Model	Volumetric factor \(f_v^{ik}\)
PC-SAFT	\(d_{ik}^3\, g_{ik}^{seg}\)
CPA	\(b_{ik}\, g(\rho)\)