Table of Contents
1 One dimensional steady-state finite-volume approximation
1.1 Summary to this point
1.2 Fick’s law of diffusion
1.3 Your turn
1.4 Your turn
1.5 Your turn
1.6 Your turn
1.7 Fick’s law of diffusion of solutes in porous media
1.8 Your turn
1.8.1 Aside: Gradients in 3 dimensions
1.9 Gridblock fluxes in terms of concentrations
1.9.1 1 Total fluxes as specific fluxes
1.10 Your turn
1.11 Your turn
1.11.1 2 Specific fluxes in terms of Fick’s law
1.11.2 3 Discrete approximation for Fick’s law
1.11.3 What is a reasonable way to do this?
1.12 Putting it all together
1.13 Reflection
One dimensional steady-state finite-volume approximation¶
Summary to this point¶
We have developed a 1-D stencil for a gridblock C to generate an steady-state equation for each interior gridblock: \begin{align*} \left(J_{EC}+J_{WC}\right) &= 0 \\ \end{align*}
where the gridblocks are sized \(\Delta x\), \(\Delta y\), \(\Delta z\)
We use the stencil to generate the equations for a simple 5 - gridblock example:
We generated the following equations:
Gridblock |
Equation |
|---|---|
1 |
\[c_1 = 2000\]
|
2 |
\[\left(J_{12}+J_{32}\right) = 0\]
|
3 |
\[\left(J_{23}+J_{43}\right) = 0\]
|
4 |
\[\left(J_{34}+J_{54}\right) = 0\]
|
5 |
\[c_5 = 93\]
|
Recall, the equations for gridblock 1 and 5 came from the boundary conditions.
Next: we need to represent the fluxes such as \(J_{12}\) in terms of the dependent variable of interest - concentration. We do that with Fick’s law of diffusion.
Fick’s law of diffusion¶
Concept: solutes (dissolved substances) move from areas of high concentration to areas of low concentration. (Why?)
Intuition: * rate of diffusion is proportional to gradient in concentration * mass flows from high concentrations towards lower concentrations
Fick’s law (the x component)
\begin{align*} j_x = - D {\partial c\over \partial x}\\ \end{align*}
where * \(j_x~\left[{M\over L^2 T}\right]\) is the x-component of the specific mass flux, * \(D\) is the diffusion coefficient and * \(\partial c\over \partial x\) is the x-component of the gradient in concentration.
Your turn¶
What is the direction of the concentration gradient?
What is the direction in which solutes are diffusing?
Your answers here.
Fick’s law of diffusion of solutes in porous media¶
We have to modify Fick’s law slightly to apply porous media. We need to introduce porosity to account for the fact that diffusion only occurs in the pore space.
\begin{align*} j_x = - D {\partial (c\theta) \over \partial x}\\ \end{align*}
where * \(\theta ~\left[{\cdot}\right]\) is the porosity (dimensionless)
Your turn¶
Now let’s compute the flux. Consider the same problem as above, where the diffusion coefficient is \(D=10^{-10}~m^2/s\), and the porosity is \(\theta = 0.3\).
If the area perpendicular to this flux direction is \(4\times10^4~m^2\) (the area of the bottom of a modest tailings pond), how much mass is transported by diffusion in one day?
Recall, that the specific flux \(j\) is the mass flux of solute per unit area per unit time and that \(J=jA\), where \(A\) is the area normal to (perpendicular to) the component of flux.
Your answer here.
Aside: Gradients in 3 dimensions¶
The gradient is a vector that points in the direction that a function is increasing. In cartesian coordinates, it has \(x\), \(y\) and \(z\) components. The diffuse flux is a gradient that points in the direction that concentration is decreasing. Hence the minus sign in Fick’s law. So the diffusive flux is also a vector: \begin{align*} j_x &= - D {\partial (c\theta) \over \partial x}\\ j_y &= - D {\partial (c\theta)\over \partial y}\\ j_z &= - D {\partial (c\theta)\over \partial z}\\ \end{align*}
We’ve switched to partial derivatives only to indicate that the concentration is a function of several independent variables (\(x\), \(y\), and \(z\)). We’ll be pretty loose with our partials and non-partials (impartials??!), but it is almost always clear from the context what is meant.
Gridblock fluxes in terms of concentrations¶
We’ll do this in three steps: 1. We’ll write the total fluxes in terms of specific fluxes. 2. We’ll write specific fluxes in terms of Fick’s law. 3. We’ll introduce a discrete approximation for Fick’s law.
1 Total fluxes as specific fluxes¶
Let’s look at our stencil equation:
First, let’s express the total fluxes in terms of specific fluxes:
Your turn¶
What is the correct value of the area \(A\) to write \(J_{WC}\) in terms of \(j_{WC}\) for this example?
Your answer here:
$A= $
Your turn¶
What is the correct value of the area \(A\) to write \(J_{WC}\) in terms of \(j_{WC}\) for the general case above?
Your answer here: $ A = $
(remember to write \(\Delta\) use $\Delta $ - or don’t bother with math type.
2 Specific fluxes in terms of Fick’s law¶
For a gridblock C oriented as below, what is the appropriate component of Fick’s law?
Choose one of (erase the two that are incorrect): \begin{align*} j_x &= - D {\partial (c\theta) \over \partial x}\\ j_y &= - D {\partial (c\theta)\over \partial y}\\ j_z &= - D {\partial (c\theta)\over \partial z}\\ \end{align*}
3 Discrete approximation for Fick’s law¶
We need a way to approximate specific fluxes such as \(j_x = - D {\partial (c\theta) \over \partial x}\) in terms of concentrations at nodal values in the centre of gridblocks as shown below.
What is a reasonable way to do this?¶
Describe in words how you would approach this.
Putting it all together¶
Can you use the three steps above to re-write the stencil
In terms of concentration? We’ll reveal the answer next…
Reflection¶
What have you learned? What have you struggled with? Some thoughts below. Reflect on anything else you have learned or struggled with and write it here.