Mode: MISCIBLEΒΆ

The miscible mode applies to a mixture of water and proplyene glycol (PPG). In terms of molar density for the mixture \(\eta\) and mole fractions \(x_i\), \(i\)=1 (water), \(i\)=2 (PPG), the mass conservation equations have the form

(1)\[\frac{{{\partial}}}{{{\partial}}t} \varphi \eta x_i + {\boldsymbol{\nabla}}\cdot\left[{\boldsymbol{q}}\eta x_i - \varphi D \eta {\boldsymbol{\nabla}}x_i\right] = Q_i,\]

with source/sink term \(Q_i\). It should be noted that the mass- and mole-fraction formulations of the conservation equations are not exactly equivalent. This is due to the diffusion term which gives an extra term when transformed from the mole-fraction to mass-fraction gradient.

The molar density \(\eta\) is related to the mass density by

(2)\[\eta = W^{-1} \rho,\]

and

(3)\[W_i\eta x_i = \rho y_i.\]

It follows that

(4)\[W_i \eta {\boldsymbol{\nabla}}x_i = \rho {\boldsymbol{\nabla}}y_i + \rho y_i {\boldsymbol{\nabla}}\ln W.\]

The second term on the right-hand side is ignored.

Simple equations of state are provided for density [g/cm:math:^3], viscosity [Pa s], and diffusivity [m:math:^2/s]. The density is a function of both composition are pressrue with the form

(5)\[\begin{split}\rho(y_1,\,p) &= \rho(y_1,\,p_0) + \left.\frac{{{\partial}}\rho}{{{\partial}}p}\right|_{p=p_0} (p-p_0),\\ &= \rho(y_1,\,p_0) \big(1+\beta (p-p_0)\big),\end{split}\]

with the compressibility \(\beta(y_1)\) given by

(6)\[\begin{split}\beta &= \left.\frac{1}{\rho}\frac{{{\partial}}\rho}{{{\partial}}p}\right|_{p=p_0},\\ &= 4.49758\times 10^{-10} y_1 + 5\times 10^{-10}(1-y_1),\end{split}\]

and the mixture density at the reference pressure \(p_0\) taken as atmospheric pressure is given by

(7)\[\rho(y_1,\,p_0) = \Big(\big((0.0806 y_1 - 0.203) y_1 + 0.0873\big) y_1 + 1.0341\Big)10^3,\]

with mass fraction of water \(y_1\). The viscosity and diffusivity have the forms

(8)\[\mu(y_1) = 10^{\big(1.6743 (1-y_1) - 0.0758\big)} 10^{-3},\]

and

(9)\[D(y_1) = \Big(\big(((-4.021 y_1 + 9.1181) y_1 - 5.9703) y_1 + 0.4043\big) y_1 + 0.5687\Big) 10^{-9},\]

The mass fraction is related to mole fraction according to

(10)\[y_1 = \frac{x_1 W_{\rm H_2O}}{W},\]

where the mean formula weight \(W\) is given by

(11)\[W = x_1 W_{\rm H_2O} + x_2 W_{\rm PPG},\]

with formula weights for water and proplyene glycol equal to \(W_{\rm H_2O}\) = 18.01534 and \(W_{\rm PPG}\) = 76.09 [kg/kmol].

Global mass conservation satisfies the relation

(12)\[\frac{d}{dt}M_i = -\int{\boldsymbol{F}}_i\cdot{\boldsymbol{dS}}+ \int Q_i dV,\]

with

(13)\[M_i = \int \varphi \eta x_i dV.\]

In terms of mass fractions and mass density

(14)\[M_i^m = W_i M_i = \int \varphi \rho y_i dV.\]