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| { | |
| "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": 8, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| "0.61\n" | |
| ] | |
| } | |
| ], | |
| "source": [ | |
| "T_a = 65 #degF\n", | |
| "T_1 = 85 #degF\n", | |
| "T_2 = 74\n", | |
| "t = 2 #hours\n", | |
| "dTdt = (T_2-T_1)/t #degF\n", | |
| "\n", | |
| "K = -dTdt/(T_2-T_a)\n", | |
| "print(round(K,2))" | |
| ] | |
| }, | |
| { | |
| "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 $K$. " | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 10, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "0.6111111111111112" | |
| ] | |
| }, | |
| "execution_count": 10, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "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", | |
| " dTdt = (Temp_t2-Temp_t1)/delta_t\n", | |
| " K = -dTdt/(Temp_t2-Temp_ambient)\n", | |
| " return K\n", | |
| "\n", | |
| "measure_K(85,74,65,2)" | |
| ] | |
| }, | |
| { | |
| "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": 3, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| "The final temperature apporaches 65 degF as time goes to infinity.\n" | |
| ] | |
| }, | |
| { | |
| "data": { | |
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\n", | |
| "text/plain": [ | |
| "<Figure size 432x288 with 1 Axes>" | |
| ] | |
| }, | |
| "metadata": { | |
| "needs_background": "light" | |
| }, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "import math\n", | |
| "import numpy as np\n", | |
| "import matplotlib.pyplot as plt\n", | |
| "\n", | |
| "#a & b\n", | |
| "\n", | |
| "t = np.linspace(0,15,35)\n", | |
| "T_o = 85 #degF\n", | |
| "T = 74 #degF\n", | |
| "T_a = 65 #degF\n", | |
| "\n", | |
| "t_numerical = np.zeros(len(t))\n", | |
| "t_numerical[0] = T_o\n", | |
| "\n", | |
| "for i in range(1,len(t)):\n", | |
| " t_numerical[i]=t_numerical[i-1]-(K*(t_numerical[i-1]-T_a)*(t[i]-t[i-1]))\n", | |
| " \n", | |
| "t_analytical = T_a + (T_o-T_a)*(np.exp(-K*t))\n", | |
| "\n", | |
| "plt.plot(t,t_numerical,label='numerical')\n", | |
| "plt.plot(t,t_analytical,label='analytical')\n", | |
| "plt.legend(loc='best')\n", | |
| "\n", | |
| "print('The final temperature apporaches 65 degF as time goes to infinity.')" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 4, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| "0 hours, 50 minutes, and 56 seconds passed since time of death.\n", | |
| "Time of death: 10:09:04 AM\n" | |
| ] | |
| } | |
| ], | |
| "source": [ | |
| "#c\n", | |
| "T_i = 98.6 #degF\n", | |
| "time = np.log((T_i-T_a)/(T_o-T_a))/K*3600\n", | |
| "\n", | |
| "hrs = math.floor(time/3600)\n", | |
| "mins = math.floor((time/(3600-hrs))*60)\n", | |
| "secs = int(round(((time/(3600-hrs))*60-mins)*60,0))\n", | |
| "\n", | |
| "print(hrs, 'hours,', mins, 'minutes, and', secs, 'seconds passed since time of death.')\n", | |
| "print('Time of death: 10:09:04 AM')" | |
| ] | |
| }, | |
| { | |
| "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": 5, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "65" | |
| ] | |
| }, | |
| "execution_count": 5, | |
| "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": [ | |
| "#a\n", | |
| "def changing_temps(hrs):\n", | |
| " '''Function defining the ambient temperature as changing each hour rather than remaining at 65 degF.\n", | |
| " This function considers 11AM as the base-time associated with T=0. (i.e. 10 is T=-1, 12 is T=1, etc.)\n", | |
| " X-axis is adjusted in plot for convenience\n", | |
| " The output of the function is changing temperature against time.'''\n", | |
| " intervals = [8,9,10,11,12,13]\n", | |
| " adj_time = hrs + 3 #adjusting time intervals\n", | |
| " a_temps = [55,58,60,65,66,67]\n", | |
| " body_temp = a_temps[adj_time]\n", | |
| " \n", | |
| " plt.plot(intervals,a_temps)\n", | |
| " return body_temp\n", | |
| "\n", | |
| "changing_temps(0)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "This plot looks like an accurate representation of ambient temperature from 8am to 1pm. A better way to record the temperature is as a continuous function, rather than 6 data points." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 6, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| "0 hours, 42 minutes, and 39 seconds passed since time of death.\n", | |
| "Time of death: 10:17:21 AM\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": [ | |
| "#b\n", | |
| "import math\n", | |
| "import numpy as np\n", | |
| "import matplotlib.pyplot as plt\n", | |
| "\n", | |
| "t = np.linspace(0,15,35)\n", | |
| "T_o = 85 #degF\n", | |
| "T = 74 #degF\n", | |
| "T_ch_amb = [55,58,60,65,67]\n", | |
| "\n", | |
| "T_amb = np.zeros(len(T_ch_amb))\n", | |
| "\n", | |
| "t_numerical = np.zeros(len(t))\n", | |
| "t_numerical[0] = T_o\n", | |
| "t_analytical = np.zeros(len(T_ch_amb))\n", | |
| "\n", | |
| "for i in range(1,len(T_ch_amb)):\n", | |
| " T_a = T_ch_amb[i]\n", | |
| " t_analytical = T_a + (T_o-T_a)*np.exp(-K*t) \n", | |
| " \n", | |
| "for j in range(1, len(t)):\n", | |
| " t_numerical[j]=t_numerical[j-1]-(K*(t_numerical[j-1]-T_a)*(t[j]-t[j-1]))\n", | |
| " \n", | |
| "plt.plot(t,t_analytical,label='analytical')\n", | |
| "plt.plot(t,t_numerical,label='numerical')\n", | |
| "plt.legend(loc='best')\n", | |
| "\n", | |
| "T_i_2 = 98.6 #degF\n", | |
| "time_2 = np.log((T_i_2-60)/(T_o-60))/K*3600\n", | |
| "\n", | |
| "hrs_2 = math.floor(time_2/3600)\n", | |
| "mins_2 = math.floor((time_2/(3600-hrs_2))*60)\n", | |
| "secs_2 = int(round(((time_2/(3600-hrs_2))*60-mins_2)*60,0))\n", | |
| "print(hrs_2, 'hours,', mins_2, 'minutes, and', secs_2, 'seconds passed since time of death.')\n", | |
| "print('Time of death: 10:17:21 AM')" | |
| ] | |
| }, | |
| { | |
| "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": "3.7.3" | |
| } | |
| }, | |
| "nbformat": 4, | |
| "nbformat_minor": 4 | |
| } |