Mode: MISCIBLE

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)$$\frac{\partial }{\partial t}\varphi \eta x_i+\mathbf{\nabla }\cdot [\mathit{q}\eta x_i-\varphi D\eta \mathbf{\nabla }{x}_i]=Q_i,$$

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)$$W_i\eta \mathbf{\nabla }{x}_i=\rho \mathbf{\nabla }{y}_i+\rho y_i\mathbf{\nabla }\ln W.$$

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)$$\begin{array}{rl}\rho (y_1,p)& =\rho (y_1,p_0)+{\frac{\partial \rho }{\partial p}|}_{p=p_0}(p-p_0),\\ & =\rho (y_1,p_0)(1+\beta (p-p_0)),\end{array}$$

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

(6)$$\begin{array}{rl}\beta & ={\frac{1}{\rho }\frac{\partial \rho }{\partial p}|}_{p=p_0},\\ & =4.49758\times {10}^{-10}{y}_1+5\times {10}^{-10}(1-y_1),\end{array}$$

and the mixture density at the reference pressure $p_0$ taken as atmospheric pressure is given by

(7)$$\rho (y_1,p_0)=(((0.0806y_1-0.203)y_1+0.0873)y_1+1.0341){10}^3,$$

with mass fraction of water $y_1$. The viscosity and diffusivity have the forms

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

and

(9)$$D(y_1)=((((-4.021y_1+9.1181)y_1-5.9703)y_1+0.4043)y_1+0.5687){10}^{-9},$$

The mass fraction is related to mole fraction according to

(10)$$y_1=\frac{x_1{W}_{{\mathrm{H}}_2\mathrm{O}}}{W},$$

where the mean formula weight $W$ is given by

(11)$$W=x_1{W}_{{\mathrm{H}}_2\mathrm{O}}+x_2{W}_{\mathrm{P}\mathrm{P}\mathrm{G}},$$

with formula weights for water and proplyene glycol equal to $W_{{\mathrm{H}}_2\mathrm{O}}$ = 18.01534 and $W_{\mathrm{P}\mathrm{P}\mathrm{G}}$ = 76.09 [kg/kmol].

Global mass conservation satisfies the relation

(12)$$\frac{d}{dt}{M}_i=-\int {\mathit{F}}_i\cdot \mathit{d}\mathit{S}+\int Q_{i}dV,$$

with

(13)$$M_i=\int \varphi \eta x_{i}dV.$$

In terms of mass fractions and mass density

(14)$$M_i^m=W_i{M}_i=\int \varphi \rho y_{i}dV.$$