{ "metadata": { "name": "" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "heading", "level": 1, "metadata": {}, "source": [ "Chapter 4: Unsteady State heat Conduction " ] }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "Example 4.1 Page No.190" ] }, { "cell_type": "code", "collapsed": false, "input": [ "\n", "k=12.0 # thermal conductivity in BTU/(hr.ft.degree Rankine) \n", "c=0.1 # specific heat in BTU/(lbm.degree Rankine) \n", "D=0.025/12.0 # diameter in ft\n", "density=525.0 # density in lbm/cu.ft\n", "hc=80 # convective coefficient in BTU/(hr. sq.ft. degree Rankine)\n", "T_i=65.0 # intial temperature in degree fahrenheit\n", "T_inf=140.0 # fluid temperature in degree fahrenheit\n", "As=3.14*D**2 # surface area in sq.ft\n", "Vs=3.14*D**(0.5) # volume in cu.ft\n", "\n", "import math\n", "reciprocal_timeconstant=(hc*6)/(density*D*c)\n", "T=139\n", "t=math.log((T-T_inf)/(T_i-T_inf))/(-reciprocal_timeconstant)\n", "\n", "print\"The response time of the junction is %.1f s\",round(t*3600,2),\"s\"\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The response time of the junction is %.1f s 3.54 s\n" ] } ], "prompt_number": 10 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "Example 4.2 Page No. 193" ] }, { "cell_type": "code", "collapsed": false, "input": [ "\n", "k=236.0 # thermal conductivity in W/(m.K)\n", "Cp=896.0 # specific heat in J/(kg.K)\n", "sp_gr=2.702 # specific gravity\n", "density=2702.0 # density in kg/cu.m\n", "D=0.05 # Diameter in m\n", "L=0.60 # length in m\n", "hc=550.0 # unit surface conductance between the metal and the bath in W/(K.sq.m)\n", "\n", "import math\n", "Vs=(math.pi*D**2*L)/4.0 # Volume in cu.m\n", "As=(2*math.pi*D**2/4.0)+(math.pi*D*L) # surface area in sq.m\n", "import math\n", "Bi=(hc*Vs)/(k*As) # Biot Number\n", "T_i=50.0 # initial temperature in degree celsius\n", "T_inf=2.0 # temperature of ice water bath in degree celsius\n", "t=60.0 # time=1 minute=60 s\n", "As_=0.102 #approx value taken in book for calculating T and Q\n", "T=T_inf+(T_i-T_inf)*math.exp(-(hc*As_*t)/(density*Vs*Cp))\n", "Q=density*Vs*Cp*(T_inf-T_i)*(1-math.exp(-(hc*As_*t)/(density*Vs*Cp)))\n", "\n", "print\"(a)The temperature of aluminium is\",round(T,1),\"C\"\n", "print\"(b)The cumulative heat transferred is \",round(-Q/1000,1),\"KJ\"\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "(a)The temperature of aluminium is 16.7 C\n", "(b)The cumulative heat transferred is 94.8 KJ\n" ] } ], "prompt_number": 8 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "Example 4.3 Page No. 200" ] }, { "cell_type": "code", "collapsed": false, "input": [ "\n", "hc=30\n", "L=0.24\n", "k=1.25 #Conductivity\n", "c=890\n", "rou=550\n", "Fo=0.4 #Fourier no\n", "\n", "Bi=hc*L/k\n", "alpha=k/(rou*c)\n", "Tc=150\n", "T_inf=600\n", "T_i=50\n", "t=(L**2*Fo)/(alpha)\n", "TC1=0.82 #Centreline temprature\n", "T=0.71*(T_i-T_inf)*TC1\n", "x=0.4*L\n", "Ti=149\n", "To=492\n", "print\"Time required to reach temprature 150 is \",round(t/3600,2),\"hr\"" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "Time required to reach temprature 150 is 2.51 hr\n" ] } ], "prompt_number": 33 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "Example 4.4 Page No. 204" ] }, { "cell_type": "code", "collapsed": false, "input": [ "hc=6 #Surface Conductance\n", "D=0.105 #Orange Diameter\n", "k=0.431 #Thermal conductivity \n", "c=2000 #Specific heat of orange\n", "rou=998 #Density\n", "Fo=1.05 #Fourier no.\n", "\n", "import math\n", "Vs=math.pi*D**3/6\n", "As=math.pi*D**2\n", "Bi=hc*Vs/(k*As)\n", "Bi_=hc*(D/2)/(k)\n", "alpha=k/(rou*c)\n", "Tc=20\n", "T_inf=23\n", "T_i=4\n", "t=(Fo*(D/2.0)**2)/alpha\n", "a=Bi_**2*Fo\n", "Q=0.7*rou*c*(math.pi/6.0*(Fo**3))*(T_i-T_inf)\n", "\n", "print\"The time required is \",round(t/3600,2),\"hr\"\n", "print\"The heat transfered is\",round(Q/1000,2),\"kj\"" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The time required is 3.72 hr\n", "The heat transfered is -16090.84 kj\n" ] } ], "prompt_number": 16 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "Example 4.5 Page No.208" ] }, { "cell_type": "code", "collapsed": false, "input": [ " \n", "D=0.105 #diameter\n", "k=0.3 #Thermal conductivity \n", "c=0.41 #Specific heat \n", "sp_gr=2.1 ##Specific gravity\n", "rou_water=62.4 #Density\n", "alpha=k/(sp_gr*rou_water*c)\n", "t=3*30*24\n", "\n", "T_inf=10\n", "Ts=10\n", "T=32\n", "T_i=70\n", "dimensionless_temp=(T-T_i)/(T_inf-T_i)\n", "variable_fig4_12=0.38 #The value of x/(2*(alpha*t)**0.5) from figure 4.12\n", "x=2*math.sqrt(alpha*t)*variable_fig4_12\n", "\n", "print\"The depth of the freeze line in soil is ft\",round(x,2),\"ft\"\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The depth of the freeze line in soil is ft 2.64 ft\n" ] } ], "prompt_number": 18 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "Example 4.6 Page No.211" ] }, { "cell_type": "code", "collapsed": false, "input": [ "\n", "k_al=236\n", "p_al=2.7*1000\n", "c_al=896\n", "k_oak=0.19\n", "p_oak=0.705*1000\n", "c_oak=2390\n", "\n", "import math\n", "math.sqrt_kpc_al=math.sqrt(k_al*p_al*c_al)\n", "kpc_R=4\n", "T_Li=20\n", "T_Ri=37.3\n", "T_al=(T_Li*(math.sqrt_kpc_al)+T_Ri*math.sqrt(kpc_R))/(math.sqrt_kpc_al+math.sqrt(kpc_R))\n", "math.sqrt_kpc_oak=math.sqrt(k_oak*p_oak*c_oak)\n", "T_oak=(T_Li*(math.sqrt_kpc_oak)+T_Ri*math.sqrt(kpc_R))/(math.sqrt_kpc_oak+math.sqrt(kpc_R))\n", "\n", "print\"The temperature of aluminium is felt as \",round(T_al,2),\"C\"\n", "print\"The temperature of oak is felt as %.1f degree celsius\",round(T_oak,1),\"C\"\n", "print\"So oak will feel warmer to the touch than will the aluminium\"\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The temperature of aluminium is felt as 20.0 C\n", "The temperature of oak is felt as %.1f degree celsius 20.1 C\n", "So oak will feel warmer to the touch than will the aluminium\n" ] } ], "prompt_number": 2 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "Example 4.7 Page No.215" ] }, { "cell_type": "code", "collapsed": false, "input": [ "\n", "rou=62.46\n", "cp=0.9988\n", "k=0.345\n", "alpha=k/(rou*cp)\n", "D=2.5/12.0\n", "L=4.75/12.0\n", "\n", "Vs=math.pi*D**2*L/4\n", "As=(math.pi*D*L)+(math.pi*D**2)/2\n", "Lc=Vs/As\n", "hc=1.7\n", "Bi=hc*Lc/k\n", "t=4\n", "\n", "Fo_cylinder=alpha*t/(D/2)**2\n", "Bi_cylinder=hc*(D/2)/k\n", "reciprocal_Bi_cylinder=1/Bi_cylinder\n", "dim_T_cylinder=0.175 #The value of dimensionless temperature of cylinder from figure 4.7a at corresponding values of Fo and 1/Bi\n", "\n", "Fo_plate=alpha*t/(L/2)**2\n", "Bi_plate=hc*L/(2*k)\n", "reciprocal_Bi_plate=1/Bi_plate\n", "dim_T_plate=0.55 #The value of dimensionless temperature of infinite plate from figure 4.7a at corresponding values of Fo and 1/Bi\n", "\n", "dim_T_shortcylinder=dim_T_cylinder*dim_T_plate\n", "T_inf=30\n", "T_i=72\n", "Tc=dim_T_shortcylinder*(T_i-T_inf)+T_inf\n", "dim_Tw_cylinder=0.77 #The dimensionless temperature from figure 4.7b corresponding to the value of 1/Bi and r/R=1\n", "dim_Tw_plate=0.65 #The dimensionless temperature from figure 4.6b corresponding to the value of 1/Bi and x/L=1\n", "dim_Tw_shortcylinder=dim_Tw_cylinder*dim_Tw_plate\n", "Tw=dim_Tw_shortcylinder*(Tc-T_inf)+T_inf\n", "\n", "print\"The temperature at centre of can is %.1f degree celsius\",round(Tc,0),\"F\"\n", "print\"The bear temperature near the metal of the can is\",round(Tw,0),\"F\"\n", "\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The temperature at centre of can is %.1f degree celsius 34.0 F\n", "The bear temperature near the metal of the can is 32.0 F\n" ] } ], "prompt_number": 38 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "Example 4.8 Page No. 219" ] }, { "cell_type": "code", "collapsed": false, "input": [ "\n", "rou=7817 #Density\n", "c=461 #Specific heat \n", "k=14.4 #Thermal conductivity \n", "alpha=.387e-5\n", "L1=0.03\n", "L2=0.03\n", "L3=0.04\n", "x=0.04\n", "T_i=95 #Internal temprature \n", "T_inf=17 #Temprature at infinity\n", "\n", "L=L1/2\n", "hc=50\n", "reciprocal_Bi_plate=k/(hc*L)\n", "Tinf=0.085 #Temprature distribution for infinite plate\n", "Tsi=0.225 #Temprature distribution for semi infinite plate\n", "T=(Tinf**2)*(1-Tsi)*(T_i-T_inf)+T_inf\n", "t=350\n", "\n", "print\"At a time 3000s The temprature is \",round(T,1),\"C\"\n", "print\"From the table The time requires to reach tempratue 50C is \",t,\"s\"" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "At a time 3000s The temprature is 17.4 C\n", "From the table The time requires to reach tempratue 50C is 350 s\n" ] } ], "prompt_number": 48 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "Example 4.9 Page No.226" ] }, { "cell_type": "code", "collapsed": false, "input": [ "%matplotlib inline" ], "language": "python", "metadata": {}, "outputs": [] }, { "cell_type": "code", "collapsed": false, "input": [ "\n", "rou=0.5*1000\n", "cp=837\n", "k=0.128\n", "alpha=0.049e-5\n", "Ti=20 #Initial temprature\n", "dt=0.5*(0.05)**2/alpha\n", "\n", "p=0\n", "T0=200\n", "m=1\n", "T11=(Ti+T0)/2.0\n", "m=2\n", "T21=(Ti+Ti)/2.0\n", "m=3\n", "T31=(Ti+Ti)/2.0\n", "m=4\n", "T41=(Ti+Ti)/2.0\n", "m=5\n", "T51=(Ti+Ti)/2.0\n", "m=6\n", "T61=(Ti+Ti)/2.0\n", "\n", "p=1\n", "m=1\n", "T12=(Ti+T0)/2.0\n", "m=2\n", "T22=(Ti+T12)/2.0\n", "m=3\n", "T32=(Ti+T21)/2.0\n", "m=4\n", "T42=(Ti+T31)/2.0\n", "m=5\n", "T52=(Ti+T41)/2.0\n", "m=6\n", "T62=(Ti+T51)/2.0\n", "t=4.97\n", "print\"The time that will pass before the heat added\",t,\"hr\"\n", "\n", "import matplotlib.pyplot as plt\n", "fig = plt.figure()\n", "ax = fig.add_subplot(111)\n", "\n", "x1=[0,30]\n", "T1=[20,20]\n", "\n", "xlabel(\"x (cm)\") \n", "ylabel(\"T (C)\") \n", "plt.xlim((0,35))\n", "plt.ylim((0,250))\n", "\n", "a1=plot(x1,T1)\n", "\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The time that will pass before the heat added 4.97 hr\n" ] }, { "output_type": "display_data", "png": 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} ], "prompt_number": 1 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "Example 4.10 Page No. 231" ] }, { "cell_type": "code", "collapsed": false, "input": [ "\n", "rou=7.817*62.4 #density\n", "c=0.110\n", "k=8.32\n", "alpha=0.417e-4\n", "dx=1/12.0\n", "Fo=1\n", "\n", "dt=Fo*dx**2/alpha\n", "n=8 #Enter the number of time intervals from Saulev plot\n", "time=n*dt\n", "\n", "print\"The required time is hr\",round(time/3600,2),\"hr\"\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The required time is hr 0.37 hr\n" ] } ], "prompt_number": 42 } ], "metadata": {} } ] }