{ "metadata": { "name": "", "signature": "sha256:762c657ca63f4bd96f109e4e82b32740084456d360cd020253f718b992f3cd8d" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "heading", "level": 1, "metadata": {}, "source": [ "Chapter 4 : Conduction of heat in the unsteady state" ] }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 4.1 page : 58" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables\n", "b = 9.; # Thickness of the wall in ft\n", "A = 5.; # Area of wall \n", "k = 0.44; # Thermal conductivity in Btu/hr-ft-degF\n", "Cp = .202; # Specific heat in Btu/lbm-degF\n", "rho = 136.; # Density in lb/ft**3\n", "\n", "\n", "# Calculations and Results\n", "\n", "def derivative(f):\n", " def df(x, h=0.1e-5):\n", " return ( f(x+h/2) - f(x-h/2) )/h\n", " return df\n", "\n", "def templength(x): # Temperature function in terms of length \n", " return 90 - 80*x +16*x**2 +32*x**3 -25.6*x**4;\n", "\n", "tgo = derivative(templength)(0); # Temperature gradient at x = 0ft\n", "tgl = derivative(templength)(9./12); # Temperature gradient at x = 9/12ft\n", "\n", "qo = -k*A*tgo; # Heat entering per unit time in Btu/hr\n", "print \"Heat entering per unit time is %.2f Btu/hr \"%(qo);\n", "ql = -k*A*tgl; # Heat coming out per unit time in Btu/hr\n", "print \" Heat coming per unit time is %.2f Btu/hr \"%(ql);\n", "q3 = qo-ql; #Heat energy stored in Btu/hr\n", "print \" Heat energy stored in wall is %.2f Btu/hr \"%(q3);\n", "\n", "a = k/(rho*Cp); # Thermal diffusivity\n", "def doublederivative(y): # Derivative of tempearture with respect to length in degF/ft\n", " return -80+32*y+96*y**2-102.4*y**3;\n", "\n", "timeder0 = a*derivative(doublederivative)(0); # derivative of temperature wrt time at x = 0 in degF\n", "print \" Time derivative of temperature wrt time at x = 0ft is %.2f degF/hr\"%(timeder0);\n", "timeder1 = a*derivative(doublederivative)(9./12); # derivative of temperature wrt time at x = 9/12 in degF \n", "print \" Time derivative of temperature wrt time at x = 9/12ft is %.2f degF/hr\"%(timeder1);\n", "\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "Heat entering per unit time is 176.00 Btu/hr \n", " Heat coming per unit time is 99.44 Btu/hr \n", " Heat energy stored in wall is 76.56 Btu/hr \n", " Time derivative of temperature wrt time at x = 0ft is 0.51 degF/hr\n", " Time derivative of temperature wrt time at x = 9/12ft is 0.05 degF/hr\n" ] } ], "prompt_number": 1 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 4.2 page :60" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables\n", "b = 9.; # thickness of the wall in ft\n", "A = 5.; # area of wall in ft**2\n", "k = 0.44; # Thermal conductivity in Btu/hr-ft-degF\n", "Cp = .202; # Specific heat in Btu/lbm-degF\n", "rho = 136.; # density in lb/ft**3\n", "\n", "# Calculations and Results\n", "def derivative(f):\n", " def df(x, h=0.1e-5):\n", " return ( f(x+h/2) - f(x-h/2) )/h\n", " return df\n", "\n", "def templength(x):\n", " return 90 - 8*x-80*x**2; \n", "\n", "\n", "tgo = derivative(templength)(0); # temperature gradient at x = 0ft\n", "tgl = derivative(templength)(9./12); # temperature gradient at x = 9/12ft\n", "\n", "qo = -k*A*tgo; # Heat entering per unit time in Btu/hr\n", "print \"Heat entering per unit time is %.2f Btu/hr \"%(qo);\n", "ql = -k*A*tgl; # Heat coming out per unit time in Btu/hr\n", "print \" Heat coming per unit time is %.2f Btu/hr \"%(ql);\n", "q3 = qo-ql; #Heat energy stored in Btu/hr\n", "print \" Heat energy stored in wall is %.2f Btu/hr \"%(q3);\n", "\n", "a = k/(rho*Cp); # Thermal diffusivity in ft**2/hr\n", "def doublederivative(y): # derivative of tempearture with respect to length in degF/ft\n", " return -8-160*y;\n", "\n", "timeder0 = a*derivative(doublederivative)(0); # derivative of temperature wrt time at x = 0 in degF\n", "print \" Time derivative of temperature wrt time at x = 0ft is %.2f degF/hr\"%(timeder0);\n", "timeder1 = a*derivative(doublederivative)(9./12); # derivative of temperature wrt time at x = 9/12 in degF \n", "print \" Time derivative of temperature wrt time at x = 9/12ft is %.2f degF/hr\"%(timeder1);\n", "print \" Teperature at each part of wall decreases equally\";\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "Heat entering per unit time is 17.60 Btu/hr \n", " Heat coming per unit time is 281.60 Btu/hr \n", " Heat energy stored in wall is -264.00 Btu/hr \n", " Time derivative of temperature wrt time at x = 0ft is -2.56 degF/hr\n", " Time derivative of temperature wrt time at x = 9/12ft is -2.56 degF/hr\n", " Teperature at each part of wall decreases equally\n" ] } ], "prompt_number": 2 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 4.3 page : 65" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables\n", "t = 10.; # time elapsed in hr\n", "Ti = 70.; # tempearature of wall initially in degF\n", "Ts = 1500.; # temperature of surface when suddenly changed in degF\n", "a = 0.03; # thermal diffusivity in ft**2/hr\n", "k = 0.5; # thermal conductivity in Btu/hr-ft-degF\n", "A = 10.; # area of wall in sq ft\n", "x = 7./12; # distance from surface where tempearture is to be found in ft\n", "f = x/(2*math.sqrt(a*t)); \n", "# From gaussian error function table erf can be found\n", "errorf = 0.55; # Referred from table\n", "\n", "# Calculations and Results\n", "T = Ts+(Ti-Ts)*errorf;\n", "print \"Temperaure at a distance of 7/12ft from surface is %.1f degF \"%(T);\n", "q = -k*A*(Ti-Ts)*math.exp(-x**2/(4*a*t))/math.sqrt(t*math.pi*a); # heat flow rate at a distance\n", "qtot = -k*A*(Ti-Ts)*2*math.sqrt(t/(math.pi*a)); # total heat flowing after 10 hrs in Btu\n", "print \" Heat flowing at a distance of 7/12 ft from surface is %d Btu/hr\"%(q); \n", "print \" Total heat flow after 10hrs is %.f Btu\"%(qtot);\n", "\n", "# note : book answer are wrong." ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "Temperaure at a distance of 7/12ft from surface is 713.5 degF \n", " Heat flowing at a distance of 7/12 ft from surface is 5546 Btu/hr\n", " Total heat flow after 10hrs is 147299 Btu\n" ] } ], "prompt_number": 8 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 4.4 page :67" ] }, { "cell_type": "code", "collapsed": false, "input": [ "\n", "# Variables\n", "d = 16./12; # Diameter of sphere in ft\n", "t = 20./60; # Time elapsed in hr\n", "a = 0.31; # thermal diffusivity of steel in ft**2/hr\n", "Ti = 80.; # Temperature of steel sphere initially in degF\n", "Ts = 1200.; # Temperature of surface suddenly changed in degF\n", "\n", "# Calculations \n", "s = 4*a*t/d**2; # A parameter \n", "# From table the value of F(s) can be known \n", "Fs = 0.20; \n", "Tc = Ts+(Ti-Ts)*Fs; # Tempearture at the center of sphere in degF\n", "\n", "# Results\n", "print \"The tempearture at the center of steel sphere after 20 mins is %d degF\"%(Tc); \n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The tempearture at the center of steel sphere after 20 mins is 976 degF\n" ] } ], "prompt_number": 9 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 4.5 page : 69" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables\n", "t = 24.; # Time period of tempearture wave in hr\n", "k = 0.6; # Thermal conductivity of wall in Btu/hr-ft-degF\n", "Cp = 0.2; # Specific heat capacity of wall in Btu/lb-degF\n", "y = 110.; # specific gravity in lb/ft**3\n", "x = 8./12; # distance from surface in ft\n", "\n", "# Calculations \n", "a = k/(y*Cp); # Thermal diffusivity in ft**2/hr\n", "n = 1./t; # frequency in /hr\n", "delr = x/(2*math.sqrt(a*math.pi*n)); # Time lag in hr\n", "\n", "# Results\n", "print \"Time lag of the temperature at a point 8 in from surface is %.1f hr\"%delr;\n", "\n", "# book answer is wrong." ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "Time lag of the temperature at a point 8 in from surface is 5.6 hr\n" ] } ], "prompt_number": 11 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 4.6 page : 69" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables\n", "T1 = -15.; # Min temperature at surface in degF\n", "T2 = 25.; # Max temperature at surface in degF\n", "t = 24.; # time gap in hrs\n", "k = 1.3; # thermal conductivity in Btu/hr-ft-degF\n", "Cp = 0.4; # heat capacity in lb/ft-degF\n", "y = 126.1; # specific gravity in lb/ft**3\n", "n = 1./t; # frequency in /hr \n", "Tm = (T1+T2)/2;\n", "a = k/(y*Cp); # thermal diffusivity in ft**2\n", "\n", "# Calculations and Results\n", "x1 = 2;\n", "x2 = 6;\n", "th0 = (T1-T2)/2;\n", "th1 = th0*-math.exp(-x1*math.sqrt(math.pi*n/a)); # temperature range at 2 ft depth\n", "th2 = th0*-math.exp(-x2*math.sqrt(math.pi*n/a)); # temperature range at 6 ft depth \n", "print \"Amplitude of tempearture at 2ft deep is %.2f degF\"%(th1);\n", "print \" Amplitude of tempearture at 6ft deep is %.2f degF\"%(th2);\n", "print \" At a depth of 2ft , temperature varies from 4.78 degF to 5.22 degF and \\\n", "at a depth of 6 ft, temperature remains constant at 5 degF\"\n", "delr1 = x1/2*math.sqrt(1/(a*math.pi*n)); # time lag at 2 ft depth\n", "delr2 = x2/2*math.sqrt(1/(a*math.pi*n)); # time lag at 6 ft depth\n", "print \" Lag of temperature wave at a depth 2 ft is %.1f hr \"%(delr1);\n", "print \" Lag of temperature wave at a depth 6 ft is %.1f hr \"%(delr2);\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "Amplitude of tempearture at 2ft deep is 0.22 degF\n", " Amplitude of tempearture at 6ft deep is 0.00 degF\n", " At a depth of 2ft , temperature varies from 4.78 degF to 5.22 degF and at a depth of 6 ft, temperature remains constant at 5 degF\n", " Lag of temperature wave at a depth 2 ft is 17.2 hr \n", " Lag of temperature wave at a depth 6 ft is 51.6 hr \n" ] } ], "prompt_number": 12 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 4.7 page : 70" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables\n", "T1 = 10.; # Min temperature at surface in degF\n", "T2 = -10.; # Max temperature at surface in degF\n", "t1 = 24.;\n", "t2 = 5.; # Time gap in hrs\n", "k = 0.3; # Thermal conductivity in Btu/hr-ft-degF\n", "Cp = 0.47; # Heat capacity in lb/ft-degF\n", "y = 100.; # Specific gravity in lb/ft**3\n", "n1 = 1/t1; # Frequency in /hr \n", "Tm = (T1+T2)/2;a = k/(y*Cp); # thermal diffusivity in ft**2\n", "n = 1./t1; # Frequency in /sec\n", "x1 = 1.;\n", "x2 = 1.;\n", "\n", "# Calculations and Results # Depth in ft\n", "th0 = (T1-T2)/2;th1 = th0*math.exp(-x1*math.sqrt(math.pi*n/a)); # temperature range at 2 ft depth\n", "th2 = th0*math.exp(-x2*math.sqrt(math.pi*n/a)); # Temperature range at 6 ft depth \n", "print \"Amplitude of tempearture at 2ft deep is %.2f degF\"%(th1);\n", "delr1 = x1/2*math.sqrt(1/(a*math.pi*n)); # Time lag at 2 ft depth\n", "print \" Lag of temperature wave at a depth 2 ft is %.1f hr \"%(delr1);\n", "# To calculate the temperature at a depth of 1 ft , 5 hr after the srface temperature reaches the minimum temperature \n", "r = 3/(4*n); # Time at which minimum surface temperature occurs for the first time in hr\n", "r1 = r+5; # Time ar which temperature is to be found out in degF\n", "th3 = th0*math.exp(-x1*math.sqrt(math.pi*n/a))*math.sin(2*math.pi*r1/24-4.53);\n", "Tr = Tm+th3; # Temperature to be found out in degF\n", "print \" The temperaure at 1 ft depth is %.2f degF \"%(Tr);\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "Amplitude of tempearture at 2ft deep is 0.11 degF\n", " Lag of temperature wave at a depth 2 ft is 17.3 hr \n", " The temperaure at 1 ft depth is 0.11 degF \n" ] } ], "prompt_number": 14 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 4.8 page : 72" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables\n", "a = 0.02; # thermal diffusivity in ft**2/hr\n", "M = 4.; # the value of 4 is selected for M\n", "\n", "# Calculations and Results\n", "x = 9./12; # thickness of wall in ft\n", "delx = 1.5/12;\n", "delr = delx**2/(a*M); # at time interval the heat transfeered will change the temperature of math.sink from tb2 to tb2o\n", "print \"The time interval is to be of %.3f hr \"%(delr);\n", "\n", "t1o = 370; \n", "t2o = 435; \n", "t3o = 480; \n", "t4o = 485; \n", "t5o = 440; \n", "t6o = 360; \n", "t7o = 250;\n", " \n", "# tempetaures at different positions at wall in degF initially\n", "# we know qo = Z*delx*dely*rho*Cp(tb2'-tb2)/delr So on solving equations we get tb2' = (tb1+tb3+ta2+tc2)/4\n", "# using above formula, temperaures at different positions as shown below can be calculated in degF\n", "\n", "ta = [370, 430, 470, 473, 431, 352, 250];\n", "tb = [370, 425, 461, 462, 422, 346, 250]; \n", "tc = [370, 420, 452, 452, 413, 341, 250];\n", "td = [370, 415, 444, 442, 404, 336, 250];\n", "print \" The temperatures at different positions 0.78 hr after, are as follows \"\n", "for i in range(7):\n", " print \" The temperature at point %d is %d degF \"%(i,td[i]);\n", "\n", "\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The time interval is to be of 0.195 hr \n", " The temperatures at different positions 0.78 hr after, are as follows \n", " The temperature at point 0 is 370 degF \n", " The temperature at point 1 is 415 degF \n", " The temperature at point 2 is 444 degF \n", " The temperature at point 3 is 442 degF \n", " The temperature at point 4 is 404 degF \n", " The temperature at point 5 is 336 degF \n", " The temperature at point 6 is 250 degF \n" ] } ], "prompt_number": 15 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 4.9 page : 73" ] }, { "cell_type": "code", "collapsed": false, "input": [ "\n", "# Variables\n", "a = 0.53; # thermal diffusivity in ft**2/hr\n", "M = 4.; # the value of 4 is selected for M\n", "x = 6./12; # thickness of wall in ft\n", "delx = 2./12;\n", "delr = delx**2/(a*M); # at time interval the heat transfeered will change the temperature of math.sink from tb2 to tb2o\n", "print \"the time interval is to be of %.3f hr \"%(delr);\n", "\n", "# the temperature is consmath.tant in the whole wall initiallt 100 degF and afterwards it changes to 1000 degF. \n", "# we know qo = Z*delx*dely*rho*Cp(tb2'-tb2)/delr So on solving equations we get tb2' = (tb1+tb3+ta2+tc2)/4\n", "# using above formula we can calculate the different temperatures as given below in degF\n", "\n", "ta = [100, 550, 775, 888, 944];\n", "tb = [100, 550, 775, 888, 944];\n", "tc = [100, 550, 775, 888, 944];\n", "td = [100, 550, 775, 888, 944];\n", "print \" the temperatures at different positions 0.052 hr after, are as follows \"\n", "print \" the temperature at point a is %d degF \"%ta[4]\n", "print \" the temperature at point a is %d degF \"%tb[4]\n", "print \" the temperature at point a is %d degF \"%tc[4];\n", "print \" the temperature at point a is %d degF \"%td[4]\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "the time interval is to be of 0.013 hr \n", " the temperatures at different positions 0.052 hr after, are as follows \n", " the temperature at point a is 944 degF \n", " the temperature at point a is 944 degF \n", " the temperature at point a is 944 degF \n", " the temperature at point a is 944 degF \n" ] } ], "prompt_number": 16 } ], "metadata": {} } ] }