Case Study 2 - Solving sets of equations - Example separation train

Case 2 - Separation train v1_1.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy.optimize import fsolve"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Specify problem parameters  \n",
    "Problem specifies:  \n",
    "  - feed rate to column 1, F  \n",
    "  - mole fraction for feed to column 1, F_mf \n",
    "  - mole fraction of all product streams, D1_mf .. B2_mf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "F = 70.0\n",
    "F_mf  = np.array([0.15, 0.25, 0.40, 0.20])\n",
    "D1_mf = np.array([0.07, 0.04, 0.54, 0.35])\n",
    "B1_mf = np.array([0.18, 0.24, 0.42, 0.16])\n",
    "D2_mf = np.array([0.15, 0.10, 0.54, 0.21])\n",
    "B2_mf = np.array([0.24, 0.65, 0.10, 0.01])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Initialise parameters to be solved\n",
    "Initialise the variables for the unspecified molar flowrates and mole fractions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "D,B,D1,B1,D2,B2 = [10.0]*6\n",
    "D_mf = np.array([0.1]*4)\n",
    "B_mf = np.array([0.1]*4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Equations describing the array of distillation columns  \n",
    "- Sum of mole fractions for each stream to sum to 1.0\n",
    "- Mass balance for each of columns 1, 2 and 3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def mass_balance(x):\n",
    "    D, B, D1, B1, D2, B2 = x[0:6]\n",
    "    D_mf = np.array(x[6:10])\n",
    "    B_mf = np.array(x[10:15])\n",
    "    \n",
    "    error_1 = 1 - sum(D_mf)\n",
    "    error_2 = 1 - sum(B_mf)\n",
    "    \n",
    "    error_mf_1 = D *D_mf   + B *B_mf   - F*F_mf\n",
    "    error_mf_2 = D1*D1_mf  + B1*B1_mf  - D*D_mf\n",
    "    error_mf_3 = D2*D2_mf  + B2*B2_mf  - B*B_mf\n",
    "    \n",
    "    return np.append([error_1, error_2], [error_mf_1, error_mf_2, error_mf_3])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem solution\n",
    "\n",
    "Solve the set of equations using fsolve from the numpy library.\n",
    "Note that the variables to be solved are based through the fsolve method as a single array."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "solution = fsolve(mass_balance, np.append([D, B, D1, B1, D2, B2],[D_mf, B_mf]))\n",
    "\n",
    "D, B, D1, B1, D2, B2 = solution[0:6]\n",
    "D_mf = solution[6:10]\n",
    "B_mf = solution[10:15]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The molar flowrate and mole fractions for all streams are given below:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[1mPrimary column\u001b[0m\n",
      "stream D:\n",
      "molar flow = 43.75 mol/min\n",
      "xylene = 0.11\n",
      "styrene = 0.12\n",
      "toluene = 0.49\n",
      "benzene = 0.27\n",
      "\n",
      "stream B:\n",
      "molar flow = 26.25 mol/min\n",
      "xylene = 0.21\n",
      "styrene = 0.47\n",
      "toluene = 0.25\n",
      "benzene = 0.08\n",
      "\n",
      "\u001b[1mDistillate column\u001b[0m\n",
      "stream D1:\n",
      "molar flow = 26.25 mol/min\n",
      "xylene = 0.07\n",
      "styrene = 0.04\n",
      "toluene = 0.54\n",
      "benzene = 0.35\n",
      "\n",
      "stream B1:\n",
      "molar flow = 17.50 mol/min\n",
      "xylene = 0.18\n",
      "styrene = 0.24\n",
      "toluene = 0.42\n",
      "benzene = 0.16\n",
      "\n",
      "\u001b[1mBottoms column\u001b[0m\n",
      "stream D2:\n",
      "molar flow = 8.75 mol/min\n",
      "xylene = 0.15\n",
      "styrene = 0.10\n",
      "toluene = 0.54\n",
      "benzene = 0.21\n",
      "\n",
      "stream B2:\n",
      "molar flow = 17.50 mol/min\n",
      "xylene = 0.24\n",
      "styrene = 0.65\n",
      "toluene = 0.10\n",
      "benzene = 0.01\n"
     ]
    }
   ],
   "source": [
    "class style:\n",
    "   BOLD = '\\033[1m'\n",
    "   END = '\\033[0m'\n",
    "\n",
    "print(style.BOLD + 'Primary column' + style.END)\n",
    "print('stream D:')\n",
    "print('molar flow = %.2f mol/min' %D)\n",
    "print('xylene = %.2f' %D_mf[0])\n",
    "print('styrene = %.2f' %D_mf[1])\n",
    "print('toluene = %.2f' %D_mf[2])\n",
    "print('benzene = %.2f' %D_mf[3])\n",
    "print('')\n",
    "print('stream B:')\n",
    "print('molar flow = %.2f mol/min' %B)\n",
    "print('xylene = %.2f' %B_mf[0])\n",
    "print('styrene = %.2f' %B_mf[1])\n",
    "print('toluene = %.2f' %B_mf[2])\n",
    "print('benzene = %.2f' %B_mf[3])\n",
    "\n",
    "print('')\n",
    "print(style.BOLD + 'Distillate column' + style.END)\n",
    "print('stream D1:')\n",
    "print('molar flow = %.2f mol/min' %D1)\n",
    "print('xylene = %.2f' %D1_mf[0])\n",
    "print('styrene = %.2f' %D1_mf[1])\n",
    "print('toluene = %.2f' %D1_mf[2])\n",
    "print('benzene = %.2f' %D1_mf[3])\n",
    "print('')\n",
    "print('stream B1:')\n",
    "print('molar flow = %.2f mol/min' %B1)\n",
    "print('xylene = %.2f' %B1_mf[0])\n",
    "print('styrene = %.2f' %B1_mf[1])\n",
    "print('toluene = %.2f' %B1_mf[2])\n",
    "print('benzene = %.2f' %B1_mf[3])\n",
    "\n",
    "print('')\n",
    "print(style.BOLD + 'Bottoms column' + style.END)\n",
    "print('stream D2:')\n",
    "print('molar flow = %.2f mol/min' %D2)\n",
    "print('xylene = %.2f' %D2_mf[0])\n",
    "print('styrene = %.2f' %D2_mf[1])\n",
    "print('toluene = %.2f' %D2_mf[2])\n",
    "print('benzene = %.2f' %D2_mf[3])\n",
    "print('')\n",
    "print('stream B2:')\n",
    "print('molar flow = %.2f mol/min' %B2)\n",
    "print('xylene = %.2f' %B2_mf[0])\n",
    "print('styrene = %.2f' %B2_mf[1])\n",
    "print('toluene = %.2f' %B2_mf[2])\n",
    "print('benzene = %.2f' %B2_mf[3])"
   ]
  }
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