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-{
- "metadata": {
- "name": "",
- "signature": "sha256:918b451a759f37d8f1e9ab3a52dd145676f732e5de5664d665d95504d9bce12d"
- },
- "nbformat": 3,
- "nbformat_minor": 0,
- "worksheets": [
- {
- "cells": [
- {
- "cell_type": "heading",
- "level": 1,
- "metadata": {},
- "source": [
- "Chapter 3 : The general property balance"
- ]
- },
- {
- "cell_type": "heading",
- "level": 3,
- "metadata": {},
- "source": [
- "Example 3.1 - Page No :65\n"
- ]
- },
- {
- "cell_type": "code",
- "collapsed": false,
- "input": [
- "\n",
- "# Variables\n",
- "a = 0.0006; \t\t #[m**2] - area\n",
- "l = 0.1; \t\t\t #[m] - length\n",
- "\n",
- "# (a) using the fourier law\n",
- "deltax = 0.1; \t\t #[m] - thickness of copper block\n",
- "T2 = 100.; \t\t #[degC] - temp on one side of copper block\n",
- "T1 = 0.; \t\t\t #[degC] - temp on other side of the copper block\n",
- "k = 380.; \t\t\t #[W/mK] - thermal conductivity\n",
- "\n",
- "# Calculations\n",
- "# using the formula (q/A)*deltax = -k*(T2-T1)\n",
- "g = -k*(T2-T1)/deltax;\n",
- "print \" a) The steady state heat flux across the copper block is q/A = %5.1e J*m**-2*sec**-1 \"%(g);\n",
- "\n",
- "# (b)\n",
- "V = a*l; \t\t\t #[m**3] - volume\n",
- "# using the overall balance equation with the accumulation and generation term\n",
- "Qgen = 1.5*10**6; \t\t\t #[j*m**-3*sec**-1]\n",
- "SIx = (g*a-Qgen*V)/a;\n",
- "\n",
- "# Results\n",
- "print \" b) the flux at face 1 is %5.1e j*m**-2*sec**-1;the negative sign indicates that the \\\n",
- "\\nheat flux is from right to left negative x direction\"%(SIx);\n",
- "\n"
- ],
- "language": "python",
- "metadata": {},
- "outputs": [
- {
- "output_type": "stream",
- "stream": "stdout",
- "text": [
- " a) The steady state heat flux across the copper block is q/A = -3.8e+05 J*m**-2*sec**-1 \n",
- " b) the flux at face 1 is -5.3e+05 j*m**-2*sec**-1;the negative sign indicates that the \n",
- "heat flux is from right to left negative x direction\n"
- ]
- }
- ],
- "prompt_number": 2
- },
- {
- "cell_type": "heading",
- "level": 3,
- "metadata": {},
- "source": [
- "Example 3.2 - Page No :68\n"
- ]
- },
- {
- "cell_type": "code",
- "collapsed": false,
- "input": [
- "\n",
- "\n",
- "from sympy import *\n",
- "\n",
- "# Variables\n",
- "x = Symbol('x')\n",
- "SIx2 = -3.8*10**5; \t\t #[j*m**-2*sec**-1] - flux at x = 0.1,i.e through face2\n",
- "Qgen = 1.5*10**6; \t\t\t #[j*m**-3*sec**-1] - uniform generation in the volume\n",
- "T2 = 100+273.15; \t\t\t #[K] temperature at face 2\n",
- "x2 = 0.1; \t\t\t #[m]\n",
- "k = 380.; \t\t\t #[W/mK] - thermal conductivity\n",
- "\n",
- "# Calculations\n",
- "# using the equation der(SIx)*x = SIx+c1;where c1 is tyhe constant of integration\n",
- "c1 = (Qgen*x2)-SIx2;\n",
- "SIx = Qgen*x-c1;\n",
- "\n",
- "# Results\n",
- "print \"SIx = \",SIx\n",
- "print \" where SIx is in units of J m**-2 sec**-1 and x is in units of m\"\n",
- "\n",
- "# using the equation -k*T = der(SIx)*x**2-c1*x+c2;where c2 is the constant of integration\n",
- "c2 = -k*T2-(Qgen*(x2)**2)/2+c1*x2;\n",
- "T = -(Qgen/k)*x**2+(c1/k)*x-(c2/k);\n",
- "print \"T = \",T\n",
- "print \" where T is in units of kelvin K\"\n",
- "\n",
- "\n",
- "# Answer may vary because of rouding error."
- ],
- "language": "python",
- "metadata": {},
- "outputs": [
- {
- "output_type": "stream",
- "stream": "stdout",
- "text": [
- "SIx = 1500000.0*x - 530000.0\n",
- " where SIx is in units of J m**-2 sec**-1 and x is in units of m\n",
- "T = -3947.36842105263*x**2 + 1394.73684210526*x + 253.413157894737\n",
- " where T is in units of kelvin K\n"
- ]
- }
- ],
- "prompt_number": 5
- },
- {
- "cell_type": "heading",
- "level": 3,
- "metadata": {},
- "source": [
- "Example 3.3 - Page No :69\n"
- ]
- },
- {
- "cell_type": "code",
- "collapsed": false,
- "input": [
- "\n",
- "\n",
- "import math \n",
- "from sympy import *\n",
- "\n",
- "\n",
- "# Variables\n",
- "# given\n",
- "x = Symbol('x')\n",
- "t = Symbol('t')\n",
- "hf1 = -270.; \t\t\t #[J/sec] - heat flow at face 1\n",
- "hf2 = -228.; \t\t\t #[J/sec] - heat flow at face2\n",
- "Qgen = 1.5*10**6; \t\t #[J*m**-3*sec**-1] generation per unit volume per unit time\n",
- "v = 6*10**-5; \t\t\t #[m**3] volume\n",
- "Cp = 0.093; \t\t\t #[cal*g**-1*K**-1] heat capacity of copper\n",
- "sp = 8.91; \t\t\t #specific gravity of copper\n",
- "a = 0.0006; \t\t\t #[m**2] - area\n",
- "\n",
- "# Calculation and Results\n",
- "# (a) using the overall balance\n",
- "acc = hf1-hf2+Qgen*v;\n",
- "print \"a) the rate of accumulation is %d J/sec \"%(acc);\n",
- "\n",
- "# (b) \n",
- "SIx1 = hf1/a;\n",
- "SIx2 = hf2/a;\n",
- "x1 = 0.;\n",
- "\n",
- "# solving for the constant of integration c1 in the equation [del(p*cp*T)/delt-der(SIx)]*x = -SIx+c1;\n",
- "c1 = 0+SIx1;\n",
- "x2 = 0.1;\n",
- "g = (-(SIx2)+c1)/x2+Qgen;\n",
- "SIx = c1-(g-Qgen)*x;\n",
- "print \"SI(x) = \",\"(b)\",SIx\n",
- "\n",
- "# solving for constant of integration c3 in the equation p*cp*T = g*t+c3\n",
- "T2 = 100+273.15;\n",
- "t2 = 0;\n",
- "p = sp*10**3; \t\t\t #[kg/m**3] - density\n",
- "cp = Cp*4.1840; \t\t\t #[J*kg**-1*K**-1]\n",
- "c3 = p*cp*T2-g*t2;\n",
- "T = (g*(10**-3)/(p*cp))*t+c3/(p*cp);\n",
- "print \"Relationship between T and t at x=0.1m is T = \",T\n",
- "\n",
- "# solving for constant of integration c2 in the equation -k*T = der(SIx)*x**2-c1*x+c2\n",
- "k = 380.; \t\t\t #[w/m**1*K**1]\n",
- "x2 = 0.1;\n",
- "c2 = k*T+(3.5*10**5)*x2**2-(4.5*10**5)*x2;\n",
- "\n",
- "def T(t,x):\n",
- " return (-(3.5*10**5)*x**2+(4.5*10**5)*x+87.7*t+1.00297*10**5)/k;\n",
- "\n",
- "# at face 1;\n",
- "x1 = 0.;\n",
- "t1 = 60.; \t\t\t #[sec]\n",
- "T1 = T(t1,x1);\n",
- "print \"Temperature profile as a function of x and t is T = %.2f K, at face 1\"%T1\n",
- "\n",
- "# at face 2\n",
- "x2 = 0.1;\n",
- "t2 = 60.; \t\t\t # [sec]\n",
- "T2 = T(t2,x2);\n",
- "print \"Temperature at face 2 = %.0f K ,at face 2\"%T2"
- ],
- "language": "python",
- "metadata": {},
- "outputs": [
- {
- "output_type": "stream",
- "stream": "stdout",
- "text": [
- "a) the rate of accumulation is 48 J/sec \n",
- "SI(x) = (b) 700000.0*x - 450000.0\n",
- "Relationship between T and t at x=0.1m is T = 0.230747847543697*t + 373.15\n",
- "Temperature profile as a function of x and t is T = 277.79 K, at face 1\n",
- "Temperature at face 2 = 387 K ,at face 2\n"
- ]
- }
- ],
- "prompt_number": 6
- }
- ],
- "metadata": {}
- }
- ]
-} \ No newline at end of file
generated by cgit (git 2.34.1) at 2025-04-05 18:04:11 +0000