I have tried to set up a model for the diffusion of a gas into a liquid. The two media are next to each other and the geometry is spherical because the system should simulate the diffusion out of a gas bubble. I have made a diffusion model for the gas phase (the left compartment) and a corresponding diffusion model for the liquid phase (the right compartment). The two diffusion models are as follows: \begin{align} \frac{\partial c_\mathrm{left}}{\partial t} &= D_\mathrm{left} \cdot \frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2 \frac{\partial c_{left}}{\partial r} \right)\\ \frac{\partial c_\mathrm{right}}{\partial t} &= D_\mathrm{right} \cdot \frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2 \frac{\partial c_{right}}{\partial r} \right)\\ \end{align}

Then I have discretized the volume. So the PDEs are turned into ODEs of the form: \begin{align} \frac{\mathrm{d}c_{i,\mathrm{left}}}{\mathrm{d}t} &= D \left(\frac{c_{i+1,\mathrm{left}} - 2 \cdot c_i + c_{i-1,\mathrm{left}}}{\Delta r^2} + \frac{2}{r_{i,\mathrm{left}}} \cdot \frac{c_{i+1,\mathrm{left}} - c_{i-1,left}}{2 \cdot \Delta r} \right)\\ \frac{\mathrm{d}c_{i,\mathrm{right}}}{\mathrm{d}t} &= D \left(\frac{c_{i+1,\mathrm{right}} - 2 \cdot c_i + c_{i-1,\mathrm{right}}}{\Delta r^2} + \frac{2}{r_{i,\mathrm{right}}} \cdot \frac{c_{i+1,\mathrm{right}} - c_{i-1,\mathrm{right}}}{2 \cdot \Delta r} \right) \end{align}

When discretizing I got the following expressions: \begin{align} \frac{\mathrm{d}^2c_i}{\mathrm{d}r^2} &= \frac{c_{i+1} - 2 \cdot c_i + c_{i-1}}{\Delta r^2}\\ \frac{\mathrm{d}c_i}{\mathrm{d}r} &= \frac{c_{i+1} - c_{i-1}}{2 \cdot \Delta r} \end{align}

But how does one incorporate the following two boundary conditions at the gas-liquid-interface, where $H$ is Henry's constant? \begin{align} c_\mathrm{left,interface} &= H \cdot c_\mathrm{right,interface}\tag1\\ J_\mathrm{left,interface} &= J_\mathrm{right,interface}\tag2 \end{align}

Meaning that there is equal flux across the interface between the two compartments.

The concentration in the last compartment of the gas phase (before the interface) is approximated by linear extrapolation. This means that the concentration just before the interface is given by: $$c_{3} = \frac{c2-c1}{r2-r1} \cdot r_3 + c_1 - r_1 \frac{c_2-c_1}{r_2 - r_1}$$

In this example it is assumed that only three compartments are present on each side of the interface.

  • 3
    $\begingroup$ If you have sucsessfully incorporated the BC for a unique equation, you'll do for these two coupled models. The BCs are now part of the coefficient matrix $L$ that solves $d_t \vec{c}-L\vec{c} =0$ and they represent 2 rows of it. You cannot approximate the concentration in the last cell... this is done precisely by the BC in each side of the boundary (interface gas-liquid). Discretise the BC you have and solve for the system! $\endgroup$ – HBR Feb 23 '18 at 7:21

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