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project/.ipynb_checkpoints/01_Getting-started-project-checkpoint.ipynb
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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 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": null, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [] | ||
| }, | ||
| { | ||
| "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": 4, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "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", | ||
| " " | ||
| ] | ||
| }, | ||
| { | ||
| "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": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "4. Now that we have a working numerical model, we can look at the results if the ambient temperature is not constant i.e. T_a=f(t). ss" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "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": "3.7.5" | ||
| } | ||
| }, | ||
| "nbformat": 4, | ||
| "nbformat_minor": 4 | ||
| } |
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