lecture_transient_heat_transfer_example.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": [
    "k1 = 1.38   # W/(m*K)\n",
    "k2 = 1.3\n",
    "k3 = 1.3\n",
    "\n",
    "c1 = 880 # J/(kg*K)\n",
    "c2 = 880\n",
    "c3 = 700\n",
    "\n",
    "rho1 = 2300 # kg/m^3\n",
    "rho2 = 2400\n",
    "rho3 = 2000\n",
    "\n",
    "L1 = 0.12  # m\n",
    "L2 = 0.23\n",
    "L3 = 0.076\n",
    "\n",
    "beta1 = 1922 # degrees K\n",
    "\n",
    "h = 20 # W/(m^2*K)\n",
    "Te = 422\n",
    "\n",
    "# initial temperatures\n",
    "T_0 = np.array([300, 300, 300, 300]) # degrees K\n",
    "\n",
    "A = 0.19 # m^2\n",
    "\n",
    "delta_t = 2000  # sec\n",
    "end_time = 300000"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "K = A*np.array([[1/A, 0, 0, 0],\n",
    "                [-k1/L1, k1/L1+k2/L2, -k2/L2, 0],\n",
    "                [0, -k2/L2, k2/L2+k3/L3, -k3/L3],\n",
    "                [0, 0, -k3/L3, k3/L3+h]])\n",
    "R = np.array([beta1,\n",
    "              0,\n",
    "              0,\n",
    "              h*A*Te])\n",
    "\n",
    "C = 0.5*A*np.diag([rho1*c1*L1, rho1*c1*L1+rho2*c2*L2, rho2*c2*L2+rho3*c3*L3, rho3*c3*L3])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1922.         1571.69011048  858.94421989  623.42818647]\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
}