Ternary Diffusion with Maxwell-Stefan Equations - Multiphase Reactor Modeling using PyMRM

Consider the two-bulb diffusion experiment where bulb A contains a 50-50 molar-% $N_{2}$ and $CO_{2}$ mixture, and bulb B contains a 50-50 molar-% $N_{2}$ and $H_{2}$ mixture. The bulbs are connected by a capillary with length $\delta = 0.086~\mathrm{m}$ . We will consider the diffusion of the three gases.

Two-Bulb Diffusion Experiment

Let’s label the three components as 1: $H_{2}$ , 2: $N_{2}$ , and 3: $CO_{2}$ . The binary Maxwell-Stefan diffusivities are:
$D_{12} = 83.6 \cdot 10^{-6}~\mathrm{m^{2}~s^{-1}}$ ,
$D_{13} = 68 \cdot 10^{-6}~\mathrm{m^{2}~s^{-1}}$ ,
$D_{23} = 16.8 \cdot 10^{-6}~\mathrm{m^{2}~s^{-1}}$ .

The pressure and temperature are $P = 1.01 \cdot 10^{5}~\mathrm{Pa}$ and $T = 308~\mathrm{K}$ , respectively.

The Maxwell-Stefan equations for the first two components can be written as:

\begin{align*} \frac{dx_{1}}{dz} &= \frac{x_{1} N_{2} - x_{2} N_{1}}{c_{tot} D_{12}} + \frac{x_{1} N_{3} - x_{3} N_{1}}{c_{tot} D_{13}}, \\ \frac{dx_{2}}{dz} &= \frac{x_{2} N_{1} - x_{1} N_{2}}{c_{tot} D_{12}} + \frac{x_{2} N_{3} - x_{3} N_{2}}{c_{tot} D_{23}}. \end{align*}

(1)

Using the constraint $x_{3} = 1 - x_{1} - x_{2}$ and the bootstrap $N_{1} + N_{2} + N_{3} = 0$ , the mole fraction and molar flux of the third component can be eliminated from the set of equations.

Questions:

Implement and numerically solve the ODEs for components 1 and 2 using the boundary conditions at $z = 0$ : $x_{1} = 0$ and $x_{2} = 0.5$ . Use a shooting method to compute $N_{1}$ and $N_{2}$ from the boundary conditions at $z = \delta$ : $x_{1} = 0.5$ and $x_{2} = 0.5$ .
Note that in Python you can use SciPy’s solve_ivp to solve the ODEs and SciPy’s root (or newton from pymrm) to obtain $N_{1}$ and $N_{2}$ such that the boundary conditions at $z = \delta$ are obeyed.
Linearize the Maxwell-Stefan equations and compute the fluxes by solving the set of two linear equations.
Use the method of Toor, Steward, and Prober to solve the ternary Maxwell-Stefan diffusion problem.
Numerically solve the problem as a boundary value problem to compute the fluxes and compositions in the capillary (e.g., using the pymrm building blocks).
Consider a time-dependent problem where the concentrations in the bulbs evolve. The bulb volumes are:
$V_{A} = 77.99 \cdot 10^{-6}~\mathrm{m^{3}}$ ,
$V_{B} = 78.63 \cdot 10^{-6}~\mathrm{m^{3}}$ .
The capillary has a diameter of $2.08~\mathrm{mm}$ , so a cross-sectional area $A_{d} = 3.40 \cdot 10^{-6}~\mathrm{m^{2}}$ , and length $d = 0.086~\mathrm{m}$ (the apparatus of Duncan and Toor, 1962). Assume constant pressure is maintained in the bulbs and the ideal gas law, such that $c_{tot}$ is equal and constant in time throughout the system and $N_{tot} = 0$ . Assume pseudo-steady state in the capillary and use a linearization to compute the flux as a function of composition in the bulbs.