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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": 531,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.275"
]
},
"execution_count": 531,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"import numpy as np\n",
"\n",
"T_a = 65;\n",
"dt = 2;\n",
"T_i = 85;\n",
"T_f = 74;\n",
"dTdt = (T_f - T_i)/dt;\n",
"K = -dTdt/(T_i - T_a);\n",
"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 $K$. "
]
},
{
"cell_type": "code",
"execution_count": 532,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.275"
]
},
"execution_count": 532,
"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",
" K = -((Temp_t2 - Temp_t1)/delta_t)/(Temp_t1 - Temp_ambient);\n",
" return K\n",
"\n",
"K = measure_K(85, 74, 65, 2)\n",
"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": 533,
"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",
"def Temperature(M,N):\n",
" t = np.linspace(0,M,N)\n",
" T_i = 85\n",
" T_a = 65\n",
" K = 0.275\n",
" T_analytical = T_a + (T_i - T_a)*(np.exp(-K*t))\n",
" \n",
" T_numerical = np.zeros(len(t));\n",
" for i in range(1, len(t)):\n",
" T_numerical[0] = T_i\n",
" dt = M/N\n",
" T_numerical[i] = T_a + (T_numerical[i - 1] - T_a)*np.exp(-K*dt);\n",
" return T_analytical, T_numerical,t\n",
"T_analytical, T_numerical, t = Temperature(20,100)\n",
"\n",
"import matplotlib.pyplot as plt\n",
"%matplotlib inline\n",
"plt.plot(t, T_analytical, '*-', label='Analytical');\n",
"plt.plot(t, T_numerical, 'o-', label='Numerical');\n",
"plt.title('Temperature over time');\n",
"plt.xlabel('Time (hr)');\n",
"plt.ylabel('Temperature ($^o$F)');\n",
"plt.legend();"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"As t $\\rightarrow\\infty$, T $\\rightarrow$ 65$^o$F."
]
},
{
"cell_type": "code",
"execution_count": 534,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Time of death: 9 :0 7 AM\n"
]
}
],
"source": [
"t = -np.log((98.6 - T_a)/(T_i - T_a))/K\n",
"time = 11+t\n",
"time;\n",
"print('Time of death:',int(time),':0','{:.0f}'.format(60*(time-int(time))),'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": 535,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"60\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": [
"#4a\n",
"\n",
"T_a = np.array([55,58,60,65,66,67])\n",
"def amb_Temp(t):\n",
" i = t + 3\n",
" Temp = T_a\n",
" print(Temp[i])\n",
" return Temp\n",
" \n",
"\n",
"Temp = amb_Temp(-1)\n",
"Temp\n",
"t = np.linspace(8,13,6)\n",
"plt.plot(t, Temp, 'o-');\n",
"plt.title('Ambient temperature over time');\n",
"plt.xlabel('Time (hr)');\n",
"plt.ylabel('Ambient Temp ($^o$ F)');"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The ambient temperature vs. time graph doesn't look quite right because there is a large jump in temperature at 10:00 AM. However, without additional data points in between, there is no way to correct that."
]
},
{
"cell_type": "code",
"execution_count": 536,
"metadata": {},
"outputs": [
{
"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": [
"#4b\n",
"\n",
"import numpy as np\n",
"def Temp_nl(M,N):\n",
" t = np.linspace(0,M,N)\n",
" T_i = 85\n",
" T_a = [55,58,60,65,66,67]\n",
" K = 0.275\n",
" T_analytical = 65 + (T_i - 65)*(np.exp(-K*t))\n",
" T_num_nl = np.zeros(len(t));\n",
" for i in range(0, len(t)):\n",
" T_num_nl[0] = T_i\n",
" dt = M/N\n",
" for j in range(0, len(T_a)):\n",
" T_num_nl[i] = T_a[j - 1] + (T_num_nl[i - 1] - T_a[j - 1])*np.exp(-K*dt);\n",
" return T_analytical,T_num_nl,t\n",
"T_analytical, T_num_nl, t = Temp_nl(2,10)\n",
"\n",
"import matplotlib.pyplot as plt\n",
"%matplotlib inline\n",
"plt.plot(t, T_analytical, '*-', label='Analytical');\n",
"plt.plot(t, T_num_nl, 'o-', label='Numerical');\n",
"plt.title('Temperature over time');\n",
"plt.xlabel('Time (hr)');\n",
"plt.ylabel('Temperature ($^o$F)');\n",
"plt.legend();"
]
},
{
"cell_type": "code",
"execution_count": 537,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Time of death: 9 : 25 AM\n"
]
}
],
"source": [
"t = -np.log((98.6 - T_a[2])/(T_i - T_a[2]))/K\n",
"time = 11+t\n",
"time;\n",
"print('Time of death:',int(time),':','{:.0f}'.format(60*(time-int(time))),'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
}