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{
"metadata": {
"name": ""
},
"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",
"(a) Find the heat flux if face 1 is maintained at 0\u00b0C.\n",
"(b) If the same flux as in part (a) enters at face 2, find the flux at face 1 if there is\n",
"now a uniform generation within the volume of 1.5 X lo6 J mm3 s-l.\n",
"'''\n",
"\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": [
"# find the temperature profile \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",
"(a) Find the rate of accumulation.\n",
"(b) Find the equation for the temperature profile, and determine the temperature\n",
"at face 1 after 1 min, if at the start face 2 is at 100\u00b0C.\n",
"'''\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": {}
}
]
}
|
