{
 "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",
    "< [Bubble and Dew Points for Multicomponent Mixtures](http://nbviewer.jupyter.org/github/jckantor/CBE20255/blob/master/notebooks/07.08-Bubble-and-Dew-Points-for-Multicomponent-Mixtures.ipynb) | [Contents](toc.ipynb) | [Binary Distillation with McCabe-Thiele](http://nbviewer.jupyter.org/github/jckantor/CBE20255/blob/master/notebooks/07.10-Binary-Distillation-with-McCabe-Thiele.ipynb) ><p><a href=\"https://colab.research.google.com/github/jckantor/CBE20255/blob/master/notebooks/07.09-Isothermal-Flash-and-the-Rachford-Rice-Equation.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": {},
   "source": [
    "# Isothermal Flash and the Rachford-Rice Equation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Summary\n",
    "\n",
    "This [Jupyter notebook](http://jupyter.org/notebook.html) illustrates the use of the Rachford-Rice equation solve the material balances for an isothermal flash of an ideal mixture. The video is used 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": {},
   "source": [
    "## Derivation of the Rachford-Rice Equation\n",
    "\n",
    "The derivation of the Rachford-Rice equation is a relatively straightford application of component material balances and Raoult's law for an ideal solution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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      "text/html": [
       "\n",
       "        <iframe\n",
       "            width=\"560\"\n",
       "            height=\"315\"\n",
       "            src=\"https://www.youtube.com/embed/ACxOiXWq1SQ?rel=0\"\n",
       "            frameborder=\"0\"\n",
       "            allowfullscreen\n",
       "        ></iframe>\n",
       "        "
      ],
      "text/plain": [
       "<IPython.lib.display.YouTubeVideo at 0x10bedb240>"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.display import YouTubeVideo\n",
    "YouTubeVideo(\"ACxOiXWq1SQ\",560,315,rel=0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The quantities, definitions, and equations are summarized in the following figure.\n",
    "\n",
    "![FlashDrumFigure.png](https://raw.githubusercontent.com/jckantor/CBE20255/master/notebooks/figures/FlashDrumFigure.png?raw=true)\n",
    "\n",
    "To sketch the derivation, we begin with the overall constraint on the liquid \n",
    "phase and vapor phase mole fractions $x_1 + x_2 + \\cdots + x_N = 1$ and $y_1 + y_2 + \\cdots + y_N = 1$. Subtracting the first from the second we find\n",
    "\n",
    "$$\\sum_{n=1}^N (y_n - x_n) = 0$$\n",
    "\n",
    "This doesn't look like much, but it turns out to be the essential trick in the development. \n",
    "\n",
    "Next we need expressions for $y_n$ and $x_n$ to substitute into terms in the sum. We get these by solving the component material balance and equilibrium equations for $y_n$ and $x_n$. For each species we write a material balance\n",
    "\n",
    "$$L x_n + V y_n = F z_n$$\n",
    "\n",
    "Dividing through by the feedrate we get a parameter $\\phi = \\frac{V}{L}$ denoting the fraction of the feedstream that leaves the flash unit in the vapor stream, the remaining fraction $1-\\phi$ leaving in the liquid stream.  With this notation the material balance becomes\n",
    "\n",
    "$$(1-\\phi)x_n + \\phi y_n = z_n$$\n",
    "\n",
    "for each species.  \n",
    "\n",
    "The second equation is\n",
    "\n",
    "$$y_n = K_n x_n$$\n",
    "\n",
    "where the 'K-factor' for an ideal solution is given by Raoult's law\n",
    "\n",
    "$$K_n = \\frac{P_n^{sat}(T)}{P}$$\n",
    "\n",
    "The K-factor depends on the operating pressure and temperature of the flash unit. Solving the material balance and equilibrium equations gives\n",
    "\n",
    "$$x_n = \\frac{z_n}{1 + \\phi(K_n - 1)}$$\n",
    "\n",
    "$$y_n = \\frac{K_n z_n}{1 + \\phi(K_n - 1)}$$\n",
    "\n",
    "so that the difference $y_n - x_n$ is given by\n",
    "\n",
    "$$y_n - x_n = \\frac{(K_n - 1)z_n}{1 + \\phi(K_n - 1)}$$\n",
    "\n",
    "Substitution leads to the Rachford-Rice equation\n",
    "\n",
    "$$\\sum_{n=1}^{N} \\frac{(K_n - 1)z_n}{1 + \\phi(K_n - 1)} = 0 $$\n",
    "\n",
    "This equation can be used to solve a variety of vapor-liquid equilibrium problems as outline in the following table."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem Classification\n",
    "\n",
    "| Problem Type | z<sub>i</sub>'s | T       | P       | &phi;   | Action |\n",
    "| -----------: | :-----: | :-----: | :-----: | :-----: | :----: |\n",
    "| Bubble Point | &check; | unknown | &check; | 0       | Set x<sub>i</sub> = z<sub>i</sub>. Solve for T and y<sub>i</sub>'s |\n",
    "| Bubble Point | &check; | &check; | unknown | 0       | Set x<sub>i</sub> = z<sub>i</sub>. Solve for P and y<sub>i</sub>'s |\n",
    "| Dew Point    | &check; | unknown | &check; | 1       | Set y<sub>i</sub> = z<sub>i</sub>. Solve for T and x<sub>i</sub>'s |\n",
    "| Dew Point    | &check; | &check; | unknown | 1       | Set y<sub>i</sub> = z<sub>i</sub>. Solve for P and x<sub>i</sub>'s |\n",
    "| Isothermal Flash | &check; | &check; | &check; | unknown | Solve for &phi;, x<sub>i</sub>'s, and y<sub>i</sub>'s |\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Isothermal Flash of a Binary Mixture\n",
    "\n",
    "Problem specifications"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "A = 'acetone'\n",
    "B = 'ethanol'\n",
    "\n",
    "P = 760\n",
    "T = 65\n",
    "\n",
    "z = dict()\n",
    "z[A] = 0.6\n",
    "z[B] = 1 - z[A]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compute the K-factors for the given operating conditions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Pressure     760.00 [mmHg]\n",
      "Temperature   65.00 [deg C]\n",
      "K-factors:\n",
      "   acetone    1.338\n",
      "   ethanol    0.576\n"
     ]
    }
   ],
   "source": [
    "# Antoine's equations. T [deg C], P [mmHg]\n",
    "Psat = dict()\n",
    "Psat[A] = lambda T: 10**(7.02447 - 1161.0/(224 + T))\n",
    "Psat[B] = lambda T: 10**(8.04494 - 1554.3/(222.65 + T))\n",
    "\n",
    "# Compute K-factors\n",
    "K = dict()\n",
    "K[A] = Psat[A](T)/P\n",
    "K[B] = Psat[B](T)/P\n",
    "\n",
    "print(\"Pressure     {:6.2f} [mmHg]\".format(P))\n",
    "print(\"Temperature  {:6.2f} [deg C]\".format(T))\n",
    "print(\"K-factors:\")\n",
    "for n in Psat:\n",
    "    print(\"   {:s}  {:7.3f}\".format(n,K[n]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Rachford-Rice equation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def RR(phi):\n",
    "    return (K[A]-1)*z[A]/(1 + phi*(K[A]-1)) + (K[B]-1)*z[B]/(1 + phi*(K[B]-1))\n",
    "\n",
    "phi = np.linspace(0,1)\n",
    "plt.plot(phi,[RR(phi) for phi in phi])\n",
    "plt.xlabel('Vapor Fraction phi')\n",
    "plt.title('Rachford-Rice Equation')\n",
    "plt.grid();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finding roots of the Rachford-Rice equation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Vapor Fraction  0.2317\n",
      "Liquid Fraction 0.7683\n"
     ]
    }
   ],
   "source": [
    "from scipy.optimize import brentq\n",
    "\n",
    "phi = brentq(RR,0,1)\n",
    "\n",
    "print(\"Vapor Fraction  {:6.4f}\".format(phi))\n",
    "print(\"Liquid Fraction {:6.4f}\".format(1-phi))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compositions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Component    z[n]    x[n]    y[n]\n",
      "acetone       0.6  0.5565  0.7444\n",
      "ethanol       0.4  0.4435  0.2556\n"
     ]
    }
   ],
   "source": [
    "x = dict()\n",
    "y = dict()\n",
    "\n",
    "print(\"Component    z[n]    x[n]    y[n]\")\n",
    "\n",
    "for n in [A,B]:\n",
    "    x[n] = z[n]/(1 + phi*(K[n]-1))\n",
    "    y[n] = K[n]*x[n]\n",
    "    print(\"{:10s} {:6.4n}  {:6.4f}  {:6.4f}\".format(n,z[n],x[n],y[n]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Multicomponent Mixtures"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "P = 760\n",
    "T = 65\n",
    "\n",
    "z = dict()\n",
    "z['acetone'] = 0.6\n",
    "z['benzene'] = 0.01\n",
    "z['toluene'] = 0.01\n",
    "z['ethanol'] = 1 - sum(z.values())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Pressure     760.00 [mmHg]\n",
      "Temperature   65.00 [deg C]\n",
      "K-factors:\n",
      "   acetone    1.338\n",
      "   benzene    0.613\n",
      "   ethanol    0.576\n",
      "   toluene    0.222\n"
     ]
    }
   ],
   "source": [
    "Psat = dict()\n",
    "Psat['acetone'] = lambda T: 10**(7.02447 - 1161.0/(224 + T))\n",
    "Psat['benzene'] = lambda T: 10**(6.89272 - 1203.531/(219.888 + T))\n",
    "Psat['ethanol'] = lambda T: 10**(8.04494 - 1554.3/(222.65 + T))\n",
    "Psat['toluene'] = lambda T: 10**(6.95805 - 1346.773/(219.693 + T))\n",
    "\n",
    "K = {n : lambda P,T,n=n: Psat[n](T)/P for n in Psat}\n",
    "\n",
    "print(\"Pressure     {:6.2f} [mmHg]\".format(P))\n",
    "print(\"Temperature  {:6.2f} [deg C]\".format(T))\n",
    "print(\"K-factors:\")\n",
    "for n in K:\n",
    "    print(\"   {:s}  {:7.3f}\".format(n,K[n](P,T)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def RR(phi):\n",
    "    return sum([(K[n](P,T)-1)*z[n]/(1 + phi*(K[n](P,T)-1)) for n in K.keys()])\n",
    "\n",
    "phi = np.linspace(0,1)\n",
    "plt.plot(phi,[RR(phi) for phi in phi])\n",
    "plt.xlabel('Vapor Fraction phi')\n",
    "plt.title('Rachford-Rice Equation')\n",
    "plt.grid();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Vapor Fraction  0.2033\n",
      "Liquid Fraction 0.7967\n"
     ]
    }
   ],
   "source": [
    "from scipy.optimize import brentq\n",
    " \n",
    "phi = brentq(RR,0,1)\n",
    "\n",
    "print(\"Vapor Fraction  {:6.4f}\".format(phi))\n",
    "print(\"Liquid Fraction {:6.4f}\".format(1-phi))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Component    z[n]    x[n]    y[n]\n",
      "acetone    0.6000  0.5615  0.7511\n",
      "benzene    0.0100  0.0109  0.0067\n",
      "toluene    0.0100  0.0119  0.0026\n",
      "ethanol    0.3800  0.4158  0.2396\n"
     ]
    }
   ],
   "source": [
    "x = {n: z[n]/(1 + phi*(K[n](P,T)-1)) for n in z}\n",
    "y = {n: K[n](P,T)*z[n]/(1 + phi*(K[n](P,T)-1)) for n in z}\n",
    "\n",
    "print(\"Component    z[n]    x[n]    y[n]\")\n",
    "\n",
    "for n in z.keys():\n",
    "    print(\"{:10s} {:6.4f}  {:6.4f}  {:6.4f}\".format(n,z[n],x[n],y[n]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[Experiments](http://en.wikipedia.org/wiki/MythBusters_%282005_season%29#Bottle_Rocket_Blast-Off) suggest the bursting pressure of a 2 liter soda bottle is 150 psig. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercises"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Design of a Carbonated Beverage\n",
    "\n",
    "The purpose of carbonating beverages is to provide a positive pressure inside the package to keep out oxygen and other potential contaminants. The burst pressure of 2 liter soda bottles [has been measured to be 150 psig](http://en.wikipedia.org/wiki/MythBusters_%282005_season%29#Bottle_Rocket_Blast-Off) (approx. 10 atm). For safety, suppose you want the bottle pressure to be no more than 6 atm gauge on a hot summer day in Arizona (say 50 &deg;C, ) and yet have at least 0.5 atm of positive gauge pressure at 0 &deg;C. Assuming your beverage is a mixture of CO<sub>2</sub> and water, is it possible to meet this specification? What concentration (measured in g of CO<sub>2</sub> per g of water) would you recommend?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<!--NAVIGATION-->\n",
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