{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "## TP - Diffusion thermique\n", "Modélisation de l'expérience vue en TP\n", "On suppose que le problème est à deux dimensions et on écrit l'équation de la diffusion thermique en régime permanent sous la forme :\n", "$$\\frac{\\partial^2T}{\\partial x^2}+\\frac{\\partial^2T}{\\partial y^2}=0$$\n", "On fixe la température à 50°C en x=0, la température de l'air extérieur est de 20°C." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": true }, "outputs": [], "source": [ "%matplotlib inline\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", "\n", "\n", "N=30\n", "Te=20\n", "T=20*np.ones((N,N),float)\n", "\n", "T[0,:]=50\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "On écrit ensuite :\n", "$$\\frac{\\partial^2T}{\\partial x^2}+\\frac{\\partial^2T}{\\partial y^2}=(T_{i+1,j}+T_{i-1,j}+T_{i,j+1}+T_{i,j-1}-4T_{i,j})/(\\Delta x)^2$$\n", "Ce qui permet de calculer $T_{i,j}$ (en prenant $\\Delta x=1$):\n", "$$T_{i,j}=(T_{i+1,j}+T_{i-1,j}+T_{i,j+1}+T_{i,j-1})/4 $$\n", "Il faut respecter les conditions aux limites : continuité du flux pour toutes les surfaces en contact avec l'air, par exemple en y=0 :\n", "$$\\lambda \\frac{\\partial T}{\\partial y}= h(T(x,0)-T_e)$$\n", "Ce qui donne :\n", "$$T_{i,0}=(a*T_{i,1}+T_e)/(1+a) $$\n", "avec \n", "$$a=\\frac{\\lambda}{h\\Delta y}$$" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": true }, "outputs": [], "source": [ "a=380/18\n", "nbiter=800\n", "k=0\n", "Ta=T.copy()\n", "\n", "for k in range(nbiter):\n", " \n", " for i in range (1,N-1):\n", " for j in range (1,N-1):\n", " T[i,j]=(Ta[i-1,j]+Ta[i+1,j]+Ta[i,j-1]+Ta[i,j+1])/4\n", " T[i,0]=(a*Ta[i,1]+Te)/(1+a)\n", " T[i,-1]=(a*Ta[i,-2]+Te)/(1+a)\n", " T[-1,j]=(a*Ta[-2,j]+Te)/(1+a)\n", " k+=1\n", " Ta=T.copy()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "On affiche ensuite la répartition de température." ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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