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{
"cells": [
{
"cell_type": "heading",
"level": 1,
"metadata": {},
"source": [
"Two dimensional, Steady State Conduction"
]
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 4.1 Page 211 "
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"\n",
"import math\n",
"d = .005; \t\t\t\t\t\t\t\t\t\t#[m] Diameter of wire\n",
"k = .35; \t\t\t\t\t\t\t\t\t\t#[W/m.K] Thermal Conductivity\n",
"h = 15; \t\t\t\t\t\t\t\t\t\t#[W/m^2.K] Total coeff with Convection n Radiation\n",
"#calculations\n",
"\n",
"rcr = k/h; \t\t\t\t\t\t\t\t\t\t# [m] critical insulation radius\n",
"tcr = rcr - d/2.; \t\t\t\t\t\t\t\t\t\t# [m] critical insulation Thickness\n",
"\n",
"Rtcond = 2.302*math.log10(rcr/(d/2.))/(2*math.pi*k); #[K/W] Thermal resistance \n",
"\n",
"#Using Table 4.1 Case 7\n",
"z = .5*tcr;\n",
"D=2*rcr;\n",
"Rtcond2D = (math.acosh((D*D + d*d - 4*z*z)/(2*D*d)))/(2*math.pi*k);\n",
"#results\n",
"\n",
"print '%s %.2f %s' %(\"\\n\\n The reduction in thermal resistance of the insulation is\", Rtcond-Rtcond2D,\" K/W \");\n",
"#END"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"\n",
"\n",
" The reduction in thermal resistance of the insulation is 0.10 K/W \n"
]
}
],
"prompt_number": 1
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 4.3 Page 224"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"\n",
"import math\n",
"import numpy\n",
"from numpy import linalg\n",
"Ts = 500.; \t#[K] Temp of surface\n",
"Tsurr = 300.; \t#[K] Temp of surrounding Air\n",
"h = 10.; \t#[W/m^2.K] Heat Convection soefficient\n",
"#Support Column\n",
"delx = .25; \t#[m]\n",
"dely = .25; \t#[m]\n",
"k = 1.; \t#[W/m.K] From Table A.3, Fireclay Brick at T = 478K\n",
"#calculations\n",
"\n",
"#Applying Eqn 4.42 and 4.48\n",
"A = numpy.array([[-4, 1, 1, 0, 0, 0, 0, 0],\n",
"\t\t[2, -4, 0, 1, 0, 0, 0, 0],\n",
"\t\t[1, 0, -4, 1, 1, 0, 0, 0],\n",
"\t\t[0, 1, 2, -4, 0, 1, 0, 0],\n",
"\t\t[0,0, 1, 0, -4, 1, 1, 0],\n",
"\t\t[0, 0, 0, 1, 2, -4, 0, 1],\n",
"\t\t[0, 0, 0, 0, 2, 0, -9, 1],\n",
"\t\t[0, 0, 0, 0, 0, 2, 2, -9]]);\n",
" \n",
"C = numpy.array([[-1000], [-500], [-500], [0], [-500], [0], [-2000], [-1500]]);\n",
"\n",
"T = numpy.linalg.solve (A,C);\n",
"#results\n",
"\n",
"print '%s' %(\"\\n Temp Distribution in K = \");\n",
"print (T);\n",
"\n",
"q = 2*h*((delx/2.)*(Ts-Tsurr)+delx*(T[6]-Tsurr)+delx*(T[7]-Tsurr)/2.);\n",
"print '%s %.2f %s' %(\"\\n\\n Heat rate from column to the airstream\",q,\" W/m \");\n",
"#END"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"\n",
" Temp Distribution in K = \n",
"[[ 489.30472333]\n",
" [ 485.15381783]\n",
" [ 472.06507549]\n",
" [ 462.00582466]\n",
" [ 436.94975396]\n",
" [ 418.73932983]\n",
" [ 356.99461052]\n",
" [ 339.05198674]]"
]
},
{
"output_type": "stream",
"stream": "stdout",
"text": [
"\n",
"\n",
"\n",
" Heat rate from column to the airstream 882.60 W/m \n"
]
}
],
"prompt_number": 2
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 4.4 Page 230"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"\n",
"import math\n",
"import numpy\n",
"from numpy import linalg\n",
"#Operating Conditions\n",
"\n",
"ho = 1000; #[W/m^2.K] Heat Convection coefficient\n",
"hi = 200; #[W/m^2.K] Heat Convection coefficient\n",
"Ti = 400; #[K] Temp of Air\n",
"Tg = 1700; #[K] Temp of Gas\n",
"h = 10 ; #[W/m^2.K] Heat Convection coefficient\n",
"\n",
"A = 2*6*math.pow(10,-6) ;#[m^2] Cross section of each Channel\n",
"x = .004 ; #[m] Spacing between joints\n",
"t = .006; #[m] Thickness\n",
"k = 25; #[W/m.K] Thermal Conductivity of Blade\n",
"delx = .001 ; #[m]\n",
"dely = .001 ; #[m]\n",
"#calculations and results\n",
"\n",
"#Applying Eqn 4.42 and 4.48\n",
"A = numpy.array([[-(2+ho*delx/k), 1, 0,0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n",
" [1,-2*(2+ho*delx/k),1,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0],\n",
" [0,1,-2*(2+ho*delx/k),1,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0],\n",
" [0,0,1,-2*(2+ho*delx/k),1,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],\n",
" [0,0,0,1,-2*(2+ho*delx/k),1,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0],\n",
" [0,0,0,0,1,-(2+ho*delx/k),0,0,0,0,0,1,0,0,0,0,0,0,0,0,0],\n",
" [1,0,0,0,0,0,-4,2,0,0,0,0,1,0,0,0,0,0,0,0,0],\n",
" [0,1,0,0,0,0,1,-4,1,0,0,0,0,1,0,0,0,0,0,0,0],\n",
" [0,0,1,0,0,0,0,1,-4,1,0,0,0,0,1,0,0,0,0,0,0],\n",
" [0,0,0,1,0,0,0,0,1,-4,1,0,0,0,0,1,0,0,0,0,0],\n",
" [0,0,0,0,1,0,0,0,0,1,-4,1,0,0,0,0,1,0,0,0,0],\n",
" [0,0,0,0,0,1,0,0,0,0,2,-4,0,0,0,0,0,1,0,0,0],\n",
" [0,0,0,0,0,0,1,0,0,0,0,0,-4,2,0,0,0,0,1,0,0],\n",
" [0,0,0,0,0,0,0,1,0,0,0,0,1,-4,1,0,0,0,0,1,0],\n",
" [0,0,0,0,0,0,0,0,2,0,0,0,0,2,-2*(3+hi*delx/k),1,0,0,0,0,1],\n",
" [0,0,0,0,0,0,0,0,0,2,0,0,0,0,1,-2*(2+hi*delx/k),1,0,0,0,0],\n",
" [0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,1,-2*(2+hi*delx/k),1,0,0,0],\n",
" [0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,-(2+hi*delx/k),0,0,0],\n",
" [0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,-2,1,0],\n",
" [0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,1,-4,1],\n",
" [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,-(2+hi*delx/k)]]);\n",
" \n",
"C = numpy.array([[-ho*delx*Tg/k], \n",
" [-2*ho*delx*Tg/k],\n",
" [-2*ho*delx*Tg/k],\n",
" [-2*ho*delx*Tg/k],\n",
" [-2*ho*delx*Tg/k],\n",
" [-ho*delx*Tg/k],\n",
" [0],\n",
" [0],\n",
" [0],\n",
" [0],\n",
" [0],\n",
" [0],\n",
" [0],\n",
" [0],\n",
" [-2*hi*delx*Ti/k],\n",
" [-2*hi*delx*Ti/k],\n",
" [-2*hi*delx*Ti/k],\n",
" [-hi*delx*Ti/k],\n",
" [0],\n",
" [0],\n",
" [-hi*delx*Ti/k]]);\n",
"\n",
"T = numpy.linalg.solve (A,C);\n",
"print '%s' %(\"\\n Temp Distribution in K = \");\n",
"print (T);\n",
"q = 4*ho*((delx/2.)*(Tg-T[0])+delx*(Tg-T[1])+delx*(Tg-T[2])+ delx*(Tg-T[3])+delx*(Tg-T[4])+delx*(Tg-T[5])/2.);\n",
"print '%s %.1f %s' %(\"\\n\\n Heat rate Transfer = \" ,q,\"W/m \");\n",
"#END"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"\n",
" Temp Distribution in K = \n",
"[[ 1525.95413813]\n",
" [ 1525.27944565]\n",
" [ 1523.59609075]\n",
" [ 1521.93574674]\n",
" [ 1520.83066847]\n",
" [ 1520.45069026]\n",
" [ 1519.66699612]\n",
" [ 1518.7949547 ]\n",
" [ 1516.52842892]\n",
" [ 1514.53554374]\n",
" [ 1513.30134519]\n",
" [ 1512.88873965]\n",
" [ 1515.12393697]\n",
" [ 1513.70494809]\n",
" [ 1509.18712651]\n",
" [ 1506.37665411]\n",
" [ 1504.9504289 ]\n",
" [ 1504.50157796]\n",
" [ 1513.41885557]\n",
" [ 1511.71377418]\n",
" [ 1506.02634497]]\n",
"\n",
"\n",
" Heat rate Transfer = 3540.6 W/m \n"
]
}
],
"prompt_number": 3
}
],
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}
]
}