{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<!--NOTEBOOK_HEADER-->\n",
    "*This notebook contains course material from [CBE20255](https://jckantor.github.io/CBE20255)\n",
    "by Jeffrey Kantor (jeff at nd.edu); the content is available [on Github](https://github.com/jckantor/CBE20255.git).\n",
    "The text is released under the [CC-BY-NC-ND-4.0 license](https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode),\n",
    "and code is released under the [MIT license](https://opensource.org/licenses/MIT).*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<!--NAVIGATION-->\n",
    "< [Ethylene Oxide Flowsheet](http://nbviewer.jupyter.org/github/jckantor/CBE20255/blob/master/notebooks/04.02-Ethylene-Oxide-Flowsheet.ipynb) | [Contents](toc.ipynb) | [Unsteady-State Material Balances](http://nbviewer.jupyter.org/github/jckantor/CBE20255/blob/master/notebooks/04.04-Unsteady-State-Material-Balances.ipynb) ><p><a href=\"https://colab.research.google.com/github/jckantor/CBE20255/blob/master/notebooks/04.03-General-Mass-Balance-on-a-Single-Tank.ipynb\"><img align=\"left\" src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open in Colab\" title=\"Open in Google Colaboratory\"></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# General Mass Balance on a Single Tank"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Summary\n",
    "\n",
    "This [Jupyter notebook](http://jupyter.org/notebook.html) demonstrates the application of a mass balance to a simple water tank. This example is adapted with permission from [learnCheme.com](http://learncheme.ning.com/), a project at the University of Colorado funded by the National Science Foundation and the Shell Corporation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Problem Statement"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Draw a Diagram"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "<img src=\"https://raw.github.com/jckantor/CBE20255/master/images/water_tank.png\" width=250px>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Mass Balance"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "Using our general principles for a mass balance\n",
    "\n",
    "$\\frac{d(\\rho V)}{dt} = \\dot{m}_1 - \\dot{m}_2$\n",
    "\n",
    "which can be simplified to\n",
    "\n",
    "$\\frac{dV}{dt} = \\frac{1}{\\rho}\\left(\\dot{m}_1 - \\dot{m}_2\\right)$\n",
    "\n",
    "where the initial value is $V(0) = 1\\,\\mbox{m}^3$. This is a differential equation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Numerical Solution using `odeint`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "There are a number of numerical methods available for solving differential equations. Here we use [`odeint`](http://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.odeint.html) which is part of the [`scipy`](http://www.scipy.org/) package.  `odeint` requires a function that returns the rate of accumulation in the tank as a function of the current volume and time."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "from scipy.integrate import odeint"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Flowrates in kg/sec\n",
    "m1 = 4.0\n",
    "m2 = 2.0\n",
    "\n",
    "# Density in kg/m**3\n",
    "rho = 1000.0\n",
    "\n",
    "# Function to compute accumulation rate\n",
    "def dV(V,t): return (m1 - m2)/rho;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Next we import `odeint` from the `scipy.integrate` package, set up a grid of times at which we wish to find solution values, then call `odeint` to compute values for the solution starting with an initial condition of 1.0."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "t = np.linspace(0,1000)\n",
    "V = odeint(dV,1.0,t)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "We finish by plotting the results of the integration and comparing to the capacity of the tank."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
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My+UcAcYCa1X1pWwOmwF0F6clsNvr+1gO1BWROiJSGujqHWtMkTZvnltH+5VX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\nxhhjgmIJwxhjTFAsYRhjjAnK/wOk5UalV0ZJKwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10e27eb38>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(t,V,'b',t,2*np.ones(len(t)),'r')\n",
    "plt.xlabel('Time [sec]')\n",
    "plt.ylabel('Volume [m**3]')\n",
    "plt.legend(['Water Volume','Tank Capacity'],loc='upper left');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This same approach can be used solve systems of differential equations. For an light-hearted (but very useful) example, check out [this solution](http://wiki.scipy.org/Cookbook/Zombie_Apocalypse_ODEINT) for the [Zombie Apocalypse](http://mysite.science.uottawa.ca/rsmith43/Zombies.pdf)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Solving for the Time Required to Fill the Tank"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Now that we know how to solve the differential equation, next we create a function to compute the air volume of the tank at any given time."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Air volume in the tank at t = 100 is 0.80 m**3.\n"
     ]
    }
   ],
   "source": [
    "Vtank = 2.0\n",
    "Vinitial = 1.0\n",
    "\n",
    "def Vwater(t): \n",
    "    return odeint(dV,Vinitial,[0,t])[-1][0]\n",
    "\n",
    "def Vair(t): \n",
    "    return Vtank - Vwater(t)\n",
    "\n",
    "print(\"Air volume in the tank at t = 100 is {:4.2f} m**3.\".format(Vair(100)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "The next step is find the time at which `Vair(t)` returns a value of 0. This is [root finding](http://docs.scipy.org/doc/scipy/reference/tutorial/optimize.html#root-finding) which the function [`brentq`](http://docs.scipy.org/doc/scipy-0.13.0/reference/generated/scipy.optimize.brentq.html) will do for us."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The tank will be full at t = 500.00 seconds.\n"
     ]
    }
   ],
   "source": [
    "from scipy.optimize import brentq\n",
    "\n",
    "t_full = brentq(Vair,0,1000)\n",
    "\n",
    "print(\"The tank will be full at t = {:6.2f} seconds.\".format(t_full))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Exercise"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Suppose the tank was being used to protect against surges in water flow, and the inlet flowrate was a function of time where\n",
    "\n",
    "$\\dot{m}_1 = 4 e^{-t/500}$\n",
    "\n",
    "* Will the tank overflow?  \n",
    "* Assuming it doesn't overflow, how long would it take for the tank to return to its initial condition of being half full? To empty completely?\n",
    "* What will be the peak volume of water in the tank, and when will that occur?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<!--NAVIGATION-->\n",
    "< [Ethylene Oxide Flowsheet](http://nbviewer.jupyter.org/github/jckantor/CBE20255/blob/master/notebooks/04.02-Ethylene-Oxide-Flowsheet.ipynb) | [Contents](toc.ipynb) | [Unsteady-State Material Balances](http://nbviewer.jupyter.org/github/jckantor/CBE20255/blob/master/notebooks/04.04-Unsteady-State-Material-Balances.ipynb) ><p><a href=\"https://colab.research.google.com/github/jckantor/CBE20255/blob/master/notebooks/04.03-General-Mass-Balance-on-a-Single-Tank.ipynb\"><img align=\"left\" src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open in Colab\" title=\"Open in Google Colaboratory\"></a>"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
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  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
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   "file_extension": ".py",
   "mimetype": "text/x-python",
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   "pygments_lexer": "ipython3",
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