{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Chapter 5 : free convection"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example 5.1 Page No : 153"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The rate of heat loss is 267 W\n"
]
}
],
"source": [
"# Variables\n",
"T1 = 65. \t\t\t#C, furnace temp.\n",
"T2 = 25. \t\t\t#C, ambient temp.\n",
"h = 1.5 \t\t\t#m, height of door\n",
"w = 1. \t\t\t#m, width of door\n",
"Tf = (T1+T2)/2 \t\t\t#c, average air film temp.\n",
"\n",
"# Calculations\n",
"Pr = 0.695 \t\t\t#Prandtl no.\n",
"mu = 1.85*10**-5 \t\t\t#m**2/s, vismath.cosity\n",
"beeta = 1/(Tf+273) \t\t\t#K**-1. coefficient of volumetric expension\n",
"k = 0.028 \t\t\t#W/m C, thermal conductivity\n",
"g = 9.8 \t\t\t#m/s**2, gravitational consmath.tant\n",
"Grl = g*beeta*(T1-T2)*h**3/(mu**2) \t\t\t#Grashof no.\n",
"Ral = Grl*Pr \t\t\t#Rayleigh no.\n",
"#Nusslet no.\n",
"Nul = (0.825+(0.387*(Ral)**(1./6))/(1+(0.492/Pr)**(9./16))**(8./27))**2 \n",
"hav = Nul*k/h \t\t\t#average heat transfer coefficient\n",
"Ad = h*w \t\t\t#m**2, door area\n",
"dt = T1-T2 \t\t\t#temp. driving force\n",
"q = hav*Ad*dt \t\t\t#W,rate of heat loss\n",
"\n",
"# Results\n",
"print \"The rate of heat loss is %.0f W\"%(q)\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example 5.2 Page No : 154"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"the steady state temp. of the plate is 61.6 C\n"
]
}
],
"source": [
"# Variables\n",
"T1 = 60. \t\t\t#C, plate temp.\n",
"T2 = 25. \t\t\t#C, ambient temp.\n",
"h = 1.\n",
"w = 1. \t\t\t#m, width of door\n",
"q = 170. \t\t\t#W, rate of heat transfer\n",
"Tf = (T1+T2)/2 \t\t\t#c, average air film temp.\n",
"#Properties of air at Tf\n",
"Pr = 0.7 \t\t\t#Prandtl no.\n",
"mu = 1.85*10**-5 \t\t\t#m**2/s, vismath.cosity\n",
"beeta = 1./(Tf+273) \t\t\t#K**-1. coefficient of volumetric expension\n",
"k = 0.028 \t\t\t#W/m C, thermal conductivity\n",
"g = 9.8 \t\t\t#m/s**2, gravitational consmath.tant\n",
"\n",
"#Calculation\n",
"A = h*w \t\t\t#m**2, plate area\n",
"P = 2*(h+w) \t\t\t#m,perimeter of plate \n",
"L = A/P \t\t\t#m characteristic length\n",
"Grl = g*beeta*(T1-T2)*L**3/(mu**2) \t\t\t#Grashof no.\n",
"Ral = Grl*Pr \t\t\t#Rayleigh no.\n",
"#Nusslet no.\n",
"Nul = 0.54*(Ral)**(1./4) \t\t\t#Nusslet no.\n",
"hav = Nul*k/L \t\t\t#average heat transfer coefficient\n",
"Ts = q/(hav*A)+T2\n",
"\n",
"# Results\n",
"print \"the steady state temp. of the plate is %.1f C\"%(Ts)\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example 5.3 Page No : 156"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The required time for cooling is 2.30 hr\n"
]
}
],
"source": [
"import math \n",
"from scipy.integrate import quad \n",
"# Variables\n",
"d = 0.0254 \t\t\t#m, diameter of steel rod\n",
"l = 0.4 \t\t\t#m, length of rod\n",
"T1 = 80. \t\t\t#C, initial temp.\n",
"T2 = 30. \t\t\t#C, ambient temp.\n",
"T3 = 35. \t\t\t#c, temp. after cooling\n",
"rho = 7800. \t\t\t#kg/m**3 ,density of steel rod\n",
"cp = 0.473 \t\t\t#kj/kg C. specific heat\n",
"\n",
"#Calculation\n",
"m = math.pi/4*d**2*l*rho \t\t\t#kg. mass of cylinder\n",
"A = math.pi*d*l \t\t\t#m**2, area of cylinder\n",
"dt = T1-T2 \t\t\t#c, insmath.tanmath.taneous temp. difference\n",
"h = 1.32*(dt/d)**0.25 \t\t\t#W/m**2 C, heat transfer coefficient\n",
"\n",
"def f0(T): \n",
" return 1./(T**(5./4))\n",
"\n",
"i = quad(f0,5,50)[0]\n",
"\n",
"t = i/(3.306*A/(m*cp*10**3))\n",
"\n",
"# Results\n",
"print \"The required time for cooling is %.2f hr\"%(t/3600.)\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example 5.4 Page No : 157"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"the rate of heat loss by free convection per meter length of pipe. is 107 W\n"
]
}
],
"source": [
"import math\n",
"# Variables\n",
"id_ = 78.*10**-3 \t\t\t#m, internal diameter\n",
"od = 89.*10**-3 \t\t\t#m, outer diameter\n",
"Pg = 15. \t\t\t#kg/cm**2, gauge pressure\n",
"t = 2.*10**-2 \t\t\t#m, thickness of preformed mineral fibre\n",
"k = 0.05 \t\t\t#W/m C. thermal conductivity\n",
"Ta = 25. \t\t\t#C, ambient air temp.\n",
"Pr = 0.705 \t\t\t#Prandtl no.\n",
"#assume\n",
"Ts = 50. \t\t\t#C, skin temp.\n",
"l = 1. \t\t\t#m, length\n",
"Ti = 200.5 \t\t\t#C, initial temp.\n",
"rs = od/2+t \t\t\t#m, outer radius of insulation\n",
"ri = od/2 \t\t\t#m, inner radius of insulation\n",
"\n",
"# Calculations\n",
"Q = 2*math.pi*l*k*(Ti-Ts)/(math.log(rs/ri)) \t\t\t#W\n",
"#properties of air at taken at the mean film temp.\n",
"Tf = (Ta+Ts)/2 \t\t\t#C\n",
"mu = 1.76*10**-5 \t\t\t#m**2/s. vismath.cosity\n",
"beeta = (1/(Tf+273)) \t\t\t#K**-1, coefficient of volumetric expansion\n",
"k1 = 0.027 \t\t\t#W/m C, thermal conductivity\n",
"ds = 2*rs \t\t\t#m, outer dia. of insulated pipe\n",
"g = 9.8 \t\t\t#m/s**2, gravitational consmath.tant\n",
"Grd = g*beeta*(Ts-Ta)*ds**3/(mu**2) \t\t\t#Grashof no.\n",
"Rad = Grd*Pr \t\t\t#Rayleigh no.\n",
"#from eq. 5.9\n",
"#Nusslet no. \n",
"Nu = (0.60+(0.387*(Rad)**(1./6))/(1+(0.559/Pr)**(9./16))**(8./27))**2 \n",
"hav = Nu*k1/ds \t\t\t#W/ m**2 C, average heat transfer coefficient\n",
"Ts = (Q/(math.pi*ds*l*hav))+Ta \t\t\t#C, skin temp.\n",
"#revised calculation by assuming\n",
"Ts1 = 70. \t\t\t#C, skin temp.\n",
"#Rate of heat transfer through insulation\n",
"Q1 = 2*math.pi*l*k*(Ti-Ts1)/(math.log(rs/ri))\n",
"Tf1 = (Ta+Ts1)/2 \t\t\t#C, average aie mean film temp.\n",
"mu1 = 1.8*10**-5 \t\t\t#m**2/s. vismath.cosity\n",
"beeta1 = (1/(Tf1+273)) \t\t\t#K**-1, coefficient of volumetric expansion\n",
"k1 = 0.0275 \t\t\t#W/m C, thermal conductivity\n",
"Pr1 = 0.703 \t\t\t#Prandtl no.\n",
"Grd1 = g*beeta1*(Ts1-Ta)*ds**3/(mu1**2) \t\t\t#Grashof no.\n",
"Rad = Grd1*Pr1 \t\t\t#Rayleigh no.\n",
"#from eq. 5.9\n",
"# average heat transfer coefficient, in \t\t\t#W/ m**2 C,\n",
"hav1 = (0.60+(0.387*(Rad)**(1./6))/(1+(0.559/Pr)**(9./16))**(8./27))**2*(k1/ds)\n",
"Ts2 = (Q1/(math.pi*ds*l*hav1))+Ta\n",
"#again assume skin temp. = 74\n",
"Ts2 = 74 \t\t\t#C, assumed skin temp.\n",
"Q3 = 2*math.pi*l*k*(Ti-Ts2)/(math.log(rs/ri))\n",
"\n",
"# Results\n",
"print \"the rate of heat loss by free convection per meter length of pipe. is %.0f W\"%(Q3)\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example 5.5 Page No : 159"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The required insulation thickness is 0.188 m\n"
]
}
],
"source": [
"from scipy.optimize import fsolve \n",
"import math \n",
"\n",
"# Variables\n",
"Ts = 65. \t\t\t#C, skin temp.\n",
"To = 30. \t\t\t#C, ambient temp.\n",
"Tw = 460. \t\t\t#C, wall temp.\n",
"Tf = (Ts+To)/2 \t\t\t#C,mean air film temp.\n",
"beeta = (1./(Tf+273)) \t\t\t#K**-1, coefficient of volumetric expansion\n",
"g = 9.8 \t\t\t#m/s**2, gravitational consmath.tant\n",
"mu = 1.84*10**-5 \t\t\t#m**2/s, vismath.cosity\n",
"L = 10.5 \t\t\t#m, height of converter\n",
"di = 4. \t\t\t#m,diameter of converter\n",
"Pr = 0.705 \t\t\t#Prandtl no.\n",
"k = 0.0241 \t\t\t#kcal/h m C, thermal conductivity\n",
"\n",
"#Calculation\n",
"Grl = g*beeta*(Ts-To)*L**3/(mu**2) \t\t\t#Grashof no.\n",
"x = di/L \t\t\t#assume di/l = x\n",
"y = 35/(Grl)**(1./4) \t\t\t#assume 35/(Grl)**(3/4) = y\n",
"#for a verticla flat plate, from eq. 5.3\n",
"Ral = Grl*Pr \t\t\t#Rayleigh no.\n",
"#nusslet no.\n",
"Nu = (0.825+(0.387*(Ral)**(1./6))/(1+(0.496/Pr)**(9./16))**(8./27))**2\n",
"hav = Nu*k/L \t\t\t#kcal/h m**2 C, average heat transfer coefficient\n",
"#w = poly(0,\"w\")\n",
"#Dav = (4+(4+2*w))/2 \t\t\t#average diameter\n",
"#Aav = math.pi*Dav*L \t\t\t#average heat transfer area\n",
"#Qi = math.pi*Dav*L*0.0602*(Tw-Ts)/w \t\t\t#Rate of heat transfer through insulation\n",
"#rate of heat transfer from the outer surface of the insulation by free convection\n",
"#Qc = hav*math.pi*Dav*L*(Ts-To) \n",
"#Qi = Qc\n",
"def f(w): \n",
" return math.pi*(4+w)*L*0.0602*(Tw-Ts)/w-hav*math.pi*(4+2*w)*L*(Ts-To)\n",
"w = fsolve(f,0.1)\n",
"\n",
"# Results\n",
"print \"The required insulation thickness is %.3f m\"%(w)\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example 5.6 Page No : 162"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"the rate of heat transfer is 13.4 W\n"
]
}
],
"source": [
"# Variables\n",
"L = 1.6 \t\t\t#m,height of enclosure\n",
"w = 0.04 \t\t\t#m, width of enclosure\n",
"b = 0.8 \t\t\t#m, breath\n",
"T1 = 22. \t\t\t#C,surface temp.\n",
"T2 = 30. \t\t\t#C, wall temp.\n",
"Tm = (T1+T2)/2 \t\t\t#C, Mean air temp.\n",
"Pr = 0.7 \t\t\t#Prandtl no.\n",
"\n",
"# Calculations\n",
"#fpr air at 26 C\n",
"beeta = 1./(Tm+273) \t\t\t#K**-1. coefficient of volumetric expension\n",
"mu = 1.684*10**-5 \t\t\t#m**2/s, vismath.cosity\n",
"k = 0.026 \t\t\t#W/m C, thermal conductivity\n",
"alpha = 2.21*10**-5 \t\t\t#m**2/s, thermal diffusity\n",
"g = 9.8 \t\t\t#m/s**2, gravitational consmath.tant\n",
"Raw = g*beeta*(T2-T1)*w**3/(mu*alpha) \t\t\t#Rayleigh no.\n",
"Nuw = 0.42*(Raw)**0.25*Pr**0.012*(L/w)**-0.3 \t\t\t#Nusslet no.\n",
"h = Nuw*k/w \t\t\t#kcal/h m**2 C, heat transfer coefficient\n",
"q = h*(T2-T1)*(L*b) \t\t\t#W,the rate of heat transfer\n",
"\n",
"# Results\n",
"print \"the rate of heat transfer is %.1f W\"%(q)\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example 5.7 Page No : 163"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"the rate of heat loss per meter length is 39.7 kcal/h\n"
]
}
],
"source": [
"import math\n",
"# Variables\n",
"Ts = 60. \t\t\t#C, surface temp\n",
"To = 30. \t\t\t#C, bulk temp.\n",
"d = 0.06 \t\t\t#m, diameter of pipe\n",
"l = 1. \t\t\t#m, length\n",
"Tm = (Ts+To)/2\n",
"#for air at Tm\n",
"rho = 1.105 \t\t\t#kg/m**3, density\n",
"cp = 0.24 \t\t\t#kcal/kg C. specific heat\n",
"mu = 1.95*10**-5 \t\t\t#kg/m s. vismath.cosity\n",
"P = 0.7 \t\t\t#Prandtl no. \n",
"kv = 1.85*10**-5 \t\t\t#m**2/s, kinetic vismath.cosity\n",
"k = 0.0241 \t\t\t#kcal/f m C, thermal conductivity\n",
"beeta = (1./(Tm+273)) \t\t\t#K**-1. coefficient of volumetric expension\n",
"V = 0.3 \t\t\t#m/s, velocity\n",
"g = 9.8 \t\t\t#m/s**2, gravitational consmath.tant\n",
"\n",
"#Calculation\n",
"Rad = g*beeta*(Ts-To)*d**3*P/(kv**2) \t\t\t#Rayleigh no.\n",
"#from eq. 5.9\n",
"Nufree = (0.60+(0.387*Rad**(1./6))/(1+(0.559/P)**(9./16))**(8./27))**2\n",
"#calculation of forced convection nusslet no.\n",
"#from eq. 4.19\n",
"Re = d*V/(kv)\n",
"Nuforced = 0.3+(0.62*Re**(1./2)*P**(1./3)/(1+(0.4/P)**(2./3))**(1./4))*(1.+(Re/(2.82*10**5))**(5./8))**(4./5)\n",
"Nu = (Nuforced**3+Nufree**3)**(1./3) \t\t\t#nusslet no. for mixed convection\n",
"#Nu = h*d/k\n",
"h = Nu*k/d \t\t\t#kcal/h m**2 C, heat transfer corfficient\n",
"q = h*math.pi*d*l*(Ts-To)\n",
"\n",
"# Results\n",
"print \"the rate of heat loss per meter length is %.1f kcal/h\"%(q)\n"
]
}
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