HW8_solution.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pylab as plt\n",
    "from math import *\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "k_c = 0.87\n",
    "k_f = 0.026\n",
    "\n",
    "c_c = 837.36 \n",
    "c_f = 1000\n",
    "\n",
    "rho_c = 2400\n",
    "rho_f = 30 \n",
    "\n",
    "T_g = 10+273\n",
    "Q_c = 100\n",
    "\n",
    "T_e = 25+273\n",
    "\n",
    "L_c = 0.1\n",
    "L_f = 0.05\n",
    "\n",
    "h = 1.3\n",
    "\n",
    "# initial temperatures\n",
    "T_0 = np.array([10+273, 10+273, 10+273]) # degrees K\n",
    "\n",
    "A = 36\n",
    "\n",
    "delta_t = 10000  # sec\n",
    "end_time = 500000"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "K = A*np.array([[1/A, 0, 0],\n",
    "                [-k_f/L_f, k_f/L_f+k_c/L_c, -k_c/L_c],\n",
    "                [0, -k_c/L_c, k_c/L_c+h]])\n",
    "\n",
    "R = np.array([T_g,\n",
    "              (A*Q_c*L_c)/2,\n",
    "              (A*Q_c*L_c)/2+A*h*T_e])\n",
    "\n",
    "\n",
    "C = 0.5*A*np.diag([rho_f*c_f*L_f, rho_f*c_f*L_f+rho_c*c_c*L_c, rho_c*c_c*L_c])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[283.         298.93882495 299.31677771]\n"
     ]
    }
   ],
   "source": [
    "# calculate steady state solution\n",
    "ss_solution = np.linalg.inv(K) @ R\n",
    "print(ss_solution)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "C_inv = np.linalg.inv(C)\n",
    "\n",
    "time = np.arange(0.0, end_time, delta_t)\n",
    "\n",
    "transient_solution = []\n",
    "\n",
    "transient_solution.append(T_0)\n",
    "\n",
    "for t in time[1:]:\n",
    "    r_bar = (C-delta_t*K) @ transient_solution[-1] + delta_t*R\n",
    "    transient_solution.append( C_inv @ r_bar)\n",
    "    \n",
    "transient_solution = np.array(transient_solution)\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Temperature vs. Time')"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot temperature of last node versus time\n",
    "plt.plot(time*(1/(60*60*24)),transient_solution[:,-1])\n",
    "plt.axhline(ss_solution[-1],linestyle='--')\n",
    "\n",
    "plt.xlabel('Time (days)')\n",
    "plt.ylabel('Temp (K)')\n",
    "plt.title('Temperature vs. Time')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "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": "3.8.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}