01_Getting-started-project.ipynb

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    "# 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, $\\frac{dT}{dt} \\approx \\frac{T(t+\\Delta t)-T(t)}{\\Delta t}$. "
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    "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 $K$. "
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    "def measure_K(Temp_t1,Temp_t2,Temp_ambient,delta_t):\n",
    "    ''' Determine the value of K based upon temperature of corpse \n",
    "    when discovered, Temp_t1\n",
    "    after time, delta_t, Temp_t2\n",
    "    with ambient temperature, Temp_ambient\n",
    "    Arguments\n",
    "    ---------\n",
    "    your inputs...\n",
    "    \n",
    "    Returns\n",
    "    -------\n",
    "    your outputs...\n",
    "    \n",
    "    '''\n",
    "    "
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    "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?"
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    "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",
    "    "
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