{ "cells": [ { "cell_type": "markdown", "metadata": { "lines_to_next_cell": 0, "toc": true }, "source": [ "

Table of Contents

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" ] }, { "cell_type": "markdown", "metadata": { "lines_to_next_cell": 0 }, "source": [ "## One dimensional steady-state finite-volume approximation\n", "\n", "### Summary to this point\n", "\n", "We have developed a 1-D stencil for a gridblock C to generate an steady-state equation for each interior gridblock:\n", "\\begin{align*}\n", "\\left(J_{EC}+J_{WC}\\right) &= 0 \\\\\n", "\\end{align*}\n", "\n", "\n", "\n", "\n", "where the gridblocks are sized $\\Delta x$, $\\Delta y$, $\\Delta z$\n", "\n", "\n", "\n", "We use the stencil to generate the equations for a simple 5 - gridblock example:\n", "\n", "\n", "\n", "\n", "We generated the following equations:\n", "\n", " | Gridblock | Equation |\n", " |:----------:|:--------------:|\n", " | 1 | $$c_1 = 2000$$|\n", " | 2 | $$\\left(J_{12}+J_{32}\\right) = 0$$|\n", " | 3 | $$\\left(J_{23}+J_{43}\\right) = 0$$|\n", " | 4 | $$\\left(J_{34}+J_{54}\\right) = 0$$|\n", " | 5 | $$c_5 = 93$$|\n", "\n", " Recall, the equations for gridblock 1 and 5 came from the boundary conditions.\n", "\n", " **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." ] }, { "cell_type": "markdown", "metadata": { "lines_to_next_cell": 0 }, "source": [ "### Fick's law of diffusion\n", "\n", "**Concept**: solutes (dissolved substances) move from areas of high concentration to areas of low concentration. (Why?)\n", "\n", "\n", "\n", "Intuition:\n", "* rate of diffusion is proportional to gradient in concentration\n", "* mass flows from high concentrations towards lower concentrations\n", "\n", "\n", "$$ flux \\propto {dc\\over dx}~\\left[{M/L^3 \\over L}\\right]$$\n", "\n", "***Fick's law*** (the x component)\n", "\n", "\\begin{align*}\n", "j_x = - D {\\partial c\\over \\partial x}\\\\\n", "\\end{align*}\n", "\n", "where\n", "* $j_x~\\left[{M\\over L^2 T}\\right]$ is the x-component of the **specific mass flux**,\n", "* $D$ is the **diffusion coefficient** and\n", "* $\\partial c\\over \\partial x$ is the x-component of the gradient in concentration.\n", "\n", "\n", "### Your turn\n", "\n", "Why the minus sign in Fick's law?\n", "\n" ] }, { "cell_type": "markdown", "metadata": { "lines_to_next_cell": 0 }, "source": [ "Your answer here" ] }, { "cell_type": "markdown", "metadata": { "lines_to_next_cell": 0 }, "source": [ "### Your turn\n", "What are the dimensions of the diffusion coefficient?" ] }, { "cell_type": "markdown", "metadata": { "lines_to_next_cell": 0 }, "source": [ "Your answer here" ] }, { "cell_type": "markdown", "metadata": { "lines_to_next_cell": 0 }, "source": [ "### Your turn\n", "\n", "What is the magnitude (value) of the concentration gradient?\n", "\n", "" ] }, { "cell_type": "markdown", "metadata": { "lines_to_next_cell": 0 }, "source": [ "Your answer here" ] }, { "cell_type": "markdown", "metadata": { "lines_to_next_cell": 0 }, "source": [ "### Your turn\n", "\n", "1. What is the direction of the concentration gradient?\n", "2. What is the direction in which solutes are diffusing?" ] }, { "cell_type": "markdown", "metadata": { "lines_to_next_cell": 0 }, "source": [ "Your answers here." ] }, { "cell_type": "markdown", "metadata": { "lines_to_next_cell": 0 }, "source": [ "\n", "### Fick's law of diffusion of solutes in porous media\n", "\n", "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.\n", "\n", "\\begin{align*}\n", "j_x = - D {\\partial (c\\theta) \\over \\partial x}\\\\\n", "\\end{align*}\n", "\n", "where\n", "* $\\theta ~\\left[{\\cdot}\\right]$ is the **porosity** (dimensionless)\n", "\n", "\n" ] }, { "cell_type": "markdown", "metadata": { "lines_to_next_cell": 0 }, "source": [ "### Your turn\n", "\n", "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 $\\theta = 0.3$.\n", "\n", "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?\n", "\n", "\n", "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.\n", "\n", "" ] }, { "cell_type": "markdown", "metadata": { "lines_to_next_cell": 0 }, "source": [ "Your answer here." ] }, { "cell_type": "markdown", "metadata": { "lines_to_next_cell": 0 }, "source": [ "#### Aside: Gradients in 3 dimensions\n", "\n", "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:\n", "\\begin{align*}\n", "j_x &= - D {\\partial (c\\theta) \\over \\partial x}\\\\\n", "j_y &= - D {\\partial (c\\theta)\\over \\partial y}\\\\\n", "j_z &= - D {\\partial (c\\theta)\\over \\partial z}\\\\\n", "\\end{align*}\n", "\n", "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.\n", "\n", "\n" ] }, { "cell_type": "markdown", "metadata": { "lines_to_next_cell": 0 }, "source": [ "### Gridblock fluxes in terms of concentrations\n", "\n", "We'll do this in three steps:\n", "1. We'll write the total fluxes in terms of specific fluxes.\n", "2. We'll write specific fluxes in terms of Fick's law.\n", "3. We'll introduce a discrete approximation for Fick's law.\n", "\n", "#### 1 Total fluxes as specific fluxes\n", "\n", "Let's look at our stencil equation:\n", "$$\n", "J_{WC}+J_{EC} =0\n", "$$\n", "\n", "First, let's express the total fluxes in terms of specific fluxes:\n", "\n", "$$J_{WC}=j_{WC}A$$\n", "\n", "### Your turn\n", "\n", "\n", "\n", "What is the correct value of the area $A$ to write $J_{WC}$ in terms of $j_{WC}$ for this example?\n" ] }, { "cell_type": "markdown", "metadata": { "lines_to_next_cell": 0 }, "source": [ "Your answer here:\n", "\n", "$A= $" ] }, { "cell_type": "markdown", "metadata": { "lines_to_next_cell": 0 }, "source": [ "### Your turn\n", "\n", "\n", "\n", "\n", "What is the correct value of the area $A$ to write $J_{WC}$ in terms of $j_{WC}$ for the general case above?\n" ] }, { "cell_type": "markdown", "metadata": { "lines_to_next_cell": 0 }, "source": [ "Your answer here:\n", "$ A = $\n", "\n", "(remember to write $\\Delta$ use `$\\Delta $` - or don't bother with math type." ] }, { "cell_type": "markdown", "metadata": { "lines_to_next_cell": 0 }, "source": [ "#### 2 Specific fluxes in terms of Fick's law\n", "\n", "For a gridblock C oriented as below, what is the appropriate component of Fick's law?\n", "\n", "\n", "\n", "Choose one of (erase the two that are incorrect):\n", "\\begin{align*}\n", "j_x &= - D {\\partial (c\\theta) \\over \\partial x}\\\\\n", "j_y &= - D {\\partial (c\\theta)\\over \\partial y}\\\\\n", "j_z &= - D {\\partial (c\\theta)\\over \\partial z}\\\\\n", "\\end{align*}\n", "\n", "\n" ] }, { "cell_type": "markdown", "metadata": { "lines_to_next_cell": 0 }, "source": [ "#### 3 Discrete approximation for Fick's law\n", "\n", "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.\n", "\n", "\n", "\n", "#### What is a reasonable way to do this?" ] }, { "cell_type": "markdown", "metadata": { "lines_to_next_cell": 0 }, "source": [ "Describe in words how you would approach this." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Putting it all together\n", "\n", "Can you use the three steps above to re-write the stencil\n", "\n", "$$\n", "J_{WC}+J_{EC} =0\n", "$$\n", "\n", "In terms of concentration? We'll reveal the answer next...\n", "\n", "\n", "### Reflection\n", "\n", "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.\n" ] } ], "metadata": { "jupytext": { "cell_metadata_filter": "all", "formats": "", "notebook_metadata_filter": "all", "text_representation": { "extension": ".py", "format_name": "percent", "format_version": "1.2", "jupytext_version": "1.0.0-rc2" } }, "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.3" }, "toc": { "base_numbering": 1, "nav_menu": {}, "number_sections": true, "sideBar": true, "skip_h1_title": false, "title_cell": "Table of Contents", "title_sidebar": "Contents", "toc_cell": true, "toc_position": {}, "toc_section_display": true, "toc_window_display": true } }, "nbformat": 4, "nbformat_minor": 2 }