01_Getting-started-project.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Computational Mechanics Project #01 - Heat Transfer in Forensic Science\n",
    "\n",
    "We can use our current skillset for a macabre application. We can predict the time of death based upon the current temperature and change in temperature of a corpse. \n",
    "\n",
    "Forensic scientists use Newton's law of cooling to determine the time elapsed since the loss of life, \n",
    "\n",
    "$\\frac{dT}{dt} = -K(T-T_a)$,\n",
    "\n",
    "where $T$ is the current temperature, $T_a$ is the ambient temperature, $t$ is the elapsed time in hours, and $K$ is an empirical constant. \n",
    "\n",
    "Suppose the temperature of the corpse is 85$^o$F at 11:00 am. Then, 2 hours later the temperature is 74$^{o}$F. \n",
    "\n",
    "Assume ambient temperature is a constant 65$^{o}$F.\n",
    "\n",
    "1. Use Python to calculate $K$ using a finite difference approximation, $\\frac{dT}{dt} \\approx \\frac{T(t+\\Delta t)-T(t)}{\\Delta t}$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 169,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.6111111111111112\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "t=np.linspace(0,120,120)\n",
    "dt = 2\n",
    "\n",
    "T1 = 85\n",
    "T2 = 74 \n",
    "Ta = 65\n",
    "\n",
    "dT = T2-T1\n",
    "\n",
    "K = -(dT/dt)/(T2-Ta)\n",
    "print(K)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2. Change your work from problem 1 to create a function that accepts the temperature at two times, ambient temperature, and the time elapsed to return $K$. \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 170,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.6111111111111112\n"
     ]
    }
   ],
   "source": [
    "def measure_K(Temp_t1,Temp_t2,Temp_ambient,delta_t):\n",
    "    import numpy as np\n",
    "\n",
    "t=np.linspace(0,120,120)\n",
    "delta_t = 2\n",
    "\n",
    "Temp_1 = 85\n",
    "Temp_2 = 74 \n",
    "T_ambient = 65\n",
    "\n",
    "dT = T2-T1\n",
    "\n",
    "\n",
    "\n",
    "K = -(dT/delta_t)/(Temp_2-T_ambient)\n",
    "print(K)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "3. A first-order thermal system has the following analytical solution, \n",
    "\n",
    "    $T(t) =T_a+(T(0)-T_a)e^{-Kt}$\n",
    "\n",
    "    where $T(0)$ is the temperature of the corpse at t=0 hours i.e. at the time of discovery and $T_a$ is a constant ambient temperature. \n",
    "\n",
    "    a. Show that an Euler integration converges to the analytical solution as the time step is decreased. Use the constant $K$ derived above and the initial temperature, T(0) = 85$^o$F. \n",
    "\n",
    "    b. What is the final temperature as t$\\rightarrow\\infty$?\n",
    "    \n",
    "    c. At what time was the corpse 98.6$^{o}$F? i.e. what was the time of death?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 171,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import math\n",
    "import matplotlib.pyplot as plt\n",
    "t=np.linspace(0,10,400)\n",
    "T_0 = 98.6\n",
    "\n",
    "T_analy = Ta + (T_0 - Ta)*np.exp(-K*t)\n",
    "plt.plot(t, T_analy)\n",
    "\n",
    "dt = 2\n",
    "T_num = np.zeros(len(t))\n",
    "\n",
    "T_num[0] = 85\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 172,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "84.96098776232222\n",
      "The initial temperature T at time = 0s is 98.6 degrees for the temperature the get to 85 degrees it took 34 mins. The time of death will be 34 mins since the corpse was found at 85 degrees 34 mins\n"
     ]
    },
    {
     "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": [
    "for i in range(1, len(t)):\n",
    "    T_num[i] = (-K*(T_num[i-1])-Ta*dt) +T_num[i-1]\n",
    "plt.plot(t, T_num, 'o')\n",
    "A = T_analy[34]\n",
    "print(A)\n",
    "\n",
    "print('The initial temperature T at time = 0s is 98.6 degrees for the temperature the get to 85 degrees it took 34 mins. The time of death will be 34 mins since the corpse was found at 85 degrees 34 mins')\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 173,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "at time goes to infinity the temperature will go to ambient temperature 65 degrees\n"
     ]
    }
   ],
   "source": [
    "print('at time goes to infinity the temperature will go to ambient temperature 65 degrees')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "4. Now that we have a working numerical model, we can look at the results if the\n",
    "ambient temperature is not constant i.e. T_a=f(t). We can use the weather to improve our estimate for time of death. Consider the following Temperature for the day in question. \n",
    "\n",
    "|time| Temp ($^o$F)|\n",
    "|---|---|\n",
    "|8am|55|\n",
    "|9am|58|\n",
    "|10am|60|\n",
    "|11am|65|\n",
    "|noon|66|\n",
    "|1pm|67|\n",
    "\n",
    "    a. Create a function that returns the current temperature based upon the time (0 hours=11am, 65$^{o}$F) \n",
    "    *Plot the function $T_a$ vs time. Does it look correct? Is there a better way to get $T_a(t)$?\n",
    "\n",
    "    b. Modify the Euler approximation solution to account for changes in temperature at each hour. \n",
    "    Compare the new nonlinear Euler approximation to the linear analytical model. \n",
    "    At what time was the corpse 98.6$^{o}$F? i.e. what was the time of death? \n",
    "    \n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 174,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f611c7988d0>]"
      ]
     },
     "execution_count": 174,
     "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": [
    "\n",
    "def time_to_temp(time):\n",
    "    \n",
    "    times = np.array([-3, -2, -1, 0, 1, 2])\n",
    "    temps = np.array([55, 58, 60, 65, 66, 67])\n",
    "    \n",
    "    if time < times[0] or time> times[-1]:\n",
    "        raise ValueError('time_temp({}) error: {} is not valid must be within the range of [-3,2]'.format(time,time))\n",
    "    else: \n",
    "        for i in range (len(times)):\n",
    "            if times[i] >= time:\n",
    "                x_1, x_2 = times[i-1], times[i]\n",
    "                y_1, y_2 = temps[i-1], temps[i]\n",
    "                return y_1 + (time - x_1)*(y_2 - y_1)/(x_2-x_1)\n",
    "            \n",
    "times = range(-3,3)\n",
    "temps = [time_to_temp(time) for time in time]\n",
    "plt.plot(time,temp)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 176,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "48 mins is the time the corpse 98.6F was found which means 48 mins before 11:00AM Which will be 10:12AM\n"
     ]
    },
    {
     "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": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "TA = []\n",
    "t = np.linspace(0, -3, 180)\n",
    "\n",
    "for i in range(len(t)):\n",
    "    TA.append(time_to_temp(t[i]))\n",
    "T_num2 = np.zeros(len(t))\n",
    "T_num2[0] = 85\n",
    "\n",
    "Ta = TA\n",
    "t = np.linspace(0, -3, 180)\n",
    "dt = 3/180 \n",
    "for i in range(1, len(t)):\n",
    "    T_num2[i] = (K*(T_num2[i-1]-Ta[i-1])*dt)+T_num2[i-1]\n",
    "    \n",
    "plt.plot(t,T_num2)\n",
    "T_num2[48]\n",
    "print('48 mins is the time the corpse 98.6F was found which means 48 mins before 11:00AM Which will be 10:12AM')"
   ]
  }
 ],
 "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": "3.7.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}