{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Chapter 4: Transient Heat Conduction" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 4.10: Heat_leaving_and_entering_the_slap.sce" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [ "//Exa 4.10\n", "clc;\n", "clear;\n", "close;\n", "//given data\n", "L=40*10^-2;// in m\n", "k=1.5;// in W/mK\n", "A=4;// in square meter\n", "alpha=1.65*10^-3;// in m^2/h\n", "//T = 50-40*x+10*x^2+20*x^3-15*x^4 , so\n", "// dtBYdx= -40+20*x+60*x^2-60*x^3\n", "// d2tBYdx2 = 20+120*x-180*x^2\n", "\n", "// Part (a) Heat entering the slab\n", "//q1= -k*A*dtBYdx , at\n", "x=0;\n", "qi= -k*A*(-40+20*x+60*x^2-60*x^3);// in w\n", "disp(qi,'Heat entering the slab in watt')\n", "// Heat leaving the slab\n", "//ql= -k*A*dtBYdx , at\n", "x=L;\n", "ql= -k*A*(-40+20*x+60*x^2-60*x^3);// in w\n", "disp(ql,'Heat leaving the slab in watt')\n", "\n", "// Part (b) Rate of heat storage\n", "RateOfHeatStorage = qi-ql;// in watt\n", "disp(RateOfHeatStorage,'Rate of heat storage in watt');\n", "\n", "// Part (c) Rate of temperature change\n", "// d2tBYdx2 = 1/alpha*dtBYdtoh\n", "// dtBYdtoh= alpha*d2tBYdx2, at\n", "x=0;\n", "dtBYdtoh = alpha*(20+120*x-180*x^2);// in degree C/h\n", "disp(dtBYdtoh,'The rate of temperature change at entering the slab in degree C/h')\n", "// dtBYdtoh= alpha*d2tBYdx2, at\n", "x=L\n", "dtBYdtoh = alpha*(20+120*x-180*x^2);// in degree C/h\n", "disp(dtBYdtoh,'The rate of temperature change at leaving the slab in degree C/h')\n", "\n", "// Part (d) for the rate of heating or cooling to be maximum\n", "// dBYdx of dtBYdtoh = 0\n", "// dBYdx of (alpha*d2tBYdx2) =0\n", "// d3tBYdx3 = 0\n", "x=120/360;// in meter\n", "disp(x,'The point where rate of heating or cooling is maximum in meter')" ] } , { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 4.11: Time_required_for_cooling_process.sce" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [ "//Exa 4.11\n", "clc;\n", "clear;\n", "close;\n", "//given data\n", "k=40;// in W/m degree C\n", "d =12*10^-3;// in meter\n", "t=127;// in degree C\n", "t_i=877;// in degree C\n", "t_infinite=52;// in degree C\n", "h= 20;// in W/m^2 degree C\n", "rho=7800;// in W/m^2K\n", "C=600;// in J/kg K\n", "r=d/2;// in meter\n", "//l_s = V/A = r/3\n", "l_s = r/3;\n", "Bi= h*l_s/k;\n", "// since Bi < 0.1 , hence lumped heat capacity analysis can be applied\n", "// (t-t_infinite)/(t_i-t_infinite) = %e^(-h*A*toh /(rho*V*C)) = %e^(-h*toh/(rho*l_s*C)) = %e^(-h*toh/(rho*C*l_s))\n", "toh = -log((t-t_infinite)/(t_i-t_infinite))*rho*C*l_s/h;// in sec\n", "disp('Time required for cooling process : '+string(toh)+' seconds or '+string(toh/60)+' minutes')" ] } , { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 4.12: Time_to_keep_furnace.sce" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [ "//Exa 4.12\n", "clc;\n", "clear;\n", "close;\n", "//given data\n", "D=10*10^-2;// in m\n", "b=D/2;\n", "h= 100;// in W/m^2 degree C\n", "T_o=418;// in degree C\n", "T_i=30;// in degree C\n", "T_infinite=1000;// in degree C\n", "\n", "disp(' (A) For copper cylinder ');\n", "k=350;// in W/mK\n", "alpha=114*10^-7;// in m^2/s\n", "Bi= h*b/k;\n", "theta_0_t = (T_o-T_infinite)/(T_i-T_infinite);\n", "Fo=18.8;\n", "// Formula Fo= alpha*t/b^2\n", "t=Fo*b^2/alpha;\n", "disp('Time required to reach for the cylinder centreline temperature 418 degree C : '+string(t)+' seconds or '+string(t/3600)+' hours')\n", "\n", "// (2) Temperature at the radius of 4 cm\n", "theta_0_t = 0.985;\n", "// Formula theta_0_t = (T-T_infinite)/(T_o-T_infinite)\n", "T= theta_0_t*(T_o-T_infinite)+T_infinite;// in degree C\n", "disp(T,'Temperature at the radius of 4 cm ') \n", "disp('It has very less temperature gradients over 4 cm radius')\n", "\n", "disp(' (B) For asbestos cylinder ');\n", "k=0.11;// in W/mK\n", "alpha=0.28*10^-7;// in m^2/s\n", "Bi= h*b/k;\n", "theta_0_t = (T_o-T_infinite)/(T_i-T_infinite);\n", "Fo=0.21;\n", "// Formula Fo= alpha*t/b^2\n", "t=Fo*b^2/alpha;\n", "disp('Time required to reach for the cylinder centreline temperature 418 degree C : '+string(t)+' seconds or '+string(t/3600)+' hours')\n", "\n", "// (2) Temperature at the radius of 4 cm\n", "theta_x_t = 0.286;\n", "// Formula theta_x_t = (T-T_infinite)/(T_o-T_infinite)\n", "T= theta_x_t*(T_o-T_infinite)+T_infinite;// in degree C\n", "disp(T,'Temperature at the radius of 4 cm ') \n", "disp('It has large temperature gradients')" ] } , { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 4.13: Centre_temperature.sce" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [ "//Exa 4.13\n", "clc;\n", "clear;\n", "close;\n", "//given data\n", "D=5*10^-2;// in m\n", "b=D/2;\n", "h= 500;// in W/m^2 degree C\n", "k=60;// in W/m^2K\n", "rho=7850;// in kg/m^3\n", "C=460;// in J/kg\n", "alpha=1.6*10^-5;// in m^2/s\n", "T_i=225;// in degree C\n", "T_infinite=25;// in degree C\n", "t=2;// in minute\n", "\n", "// Part(i)\n", "Bi= h*b/k;\n", "Fo= alpha*t/b^2;\n", "theta_0_t = 0.18;\n", "// Formula theta_0_t = (T_o-T_infinite)/(T_i-T_infinite)\n", "T_o= theta_0_t*(T_i-T_infinite)+T_infinite;// in degree C\n", "disp(T_o,'Centreline Temperature of the sphere after 2 minutes of exposure in degree C ') ;\n", "\n", "// Part(2)\n", "depth= 10*10^-3;// in meter\n", "r=b-depth;// in meter\n", "rBYb=r/b;\n", "theta_x_t = 0.95;\n", "// Formula theta_x_t = (T-T_infinite)/(T_o-T_infinite)\n", "T= theta_x_t*(T_o-T_infinite)+T_infinite;// in degree C\n", "disp(T,'The Temperature at the depth of 1 cm from the surface after 2 minutes in degree C ') ;\n", "\n", "// Part (3)\n", "BiSquareFo= Bi^2*Fo;\n", "QbyQo= 0.8;// in kJ\n", "A=4/3*%pi*b^3;\n", "Qo= rho*A*C*(T_i-T_infinite);// in J\n", "Qo=Qo*10^-3;// in kJ\n", "// The heat transffered during 2 minute, \n", "Q= Qo*QbyQo;// in kJ\n", "disp(Q,'The heat transffered during 2 minutes in kJ')\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "" ] } , { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 4.1: Rate_of_change_of_energy_storage_in_the_wall.sce" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [ "//Exa 4.1\n", "clc;\n", "clear;\n", "close;\n", "//given data\n", "format('v',9)\n", "L=1;// in m\n", "rho=1600;// in kg/m^3\n", "k=40;// in w/mK\n", "Cp=4*10^3;// in J/kgK\n", "a=900;// in degree C\n", "b=-300;// in degree C/m\n", "c=-50;// in degree C/m^2\n", "Qg=1*10^3;// in kW/m^2\n", "A=10;// area in m^2\n", "//t=a+b*x+c*x^2 at any instant, so\n", "// dtBYdx= b+2*c*x\n", "// d2tBYdx2 = 2*c, then\n", "\n", "// Part(a)\n", "//q1= -k*A*dtBYdx , at\n", "x=0;\n", "q1= -k*A*(b+2*c*x);// in w\n", "//q2= -k*A*dtBYdx , at\n", "x=L;\n", "q2= -k*A*(b+2*c*x);// in w\n", "E_stored= (q1-q2)+Qg*A*L;// in watt\n", "disp(E_stored,'The rate of change of energy storage in watt')\n", "\n", "// Part(b)\n", "alpha= k/(rho*Cp);// in m^2s\n", "d2tBYdx2 = 2*c;\n", "dtBYdtoh= alpha*(d2tBYdx2+Qg/k );// in degree C/sec\n", "disp(dtBYdtoh,'Rate of change of temperature in degree C/sec');\n", "disp('Since dt by dx is independent of x. Hence time rate of charge of temperature throughout wall will remain same.')\n", "\n", "\n", "\n", "" ] } , { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 4.2: EX4_2.sce" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [ "//Exa 4.2\n", "clc;\n", "clear;\n", "close;\n", "//given data\n", "k=40;// in W/mK\n", "rho=7800;// in kg/m^3\n", "C=450;// in J/kgK\n", "d=20*10^-3;// in m\n", "r=d/2;\n", "t_i=400;// in degree C\n", "t=85;// in degree C\n", "t_infinite=25;// in degree C\n", "h=80;// in W/m^2K\n", "//l_s=V/A = (4/3*%pi*r^3)/(4*%pi*r^2) = r/3\n", "l_s=r/3;// in m\n", "Bi= h*l_s/k;\n", "// since Biot number is less than 0.1, hence lumped heat capacity system analysis can be applied\n", "\n", "// Part(a)\n", "// Formula (t-t_infinite)/(t_i-t_infinite)= %e^(-h*A*toh/(rho*V*C)) = %e^(-h*toh/(rho*l_s*C))\n", "toh= -log((t-t_infinite)/(t_i-t_infinite))*(rho*l_s*C)/h;// in sec\n", "disp(toh,'The time require to cool the sphere in sec');\n", "\n", "// Part(b)\n", "// dtBYdtoh = h*A*(t_i-t_infinite)/(rho*V*C) = h*(t_i-t_infinite)/(rho*l_s*C)\n", "dtBYdtoh = h*(t_i-t_infinite)/(rho*l_s*C);// in degree C/sec\n", "disp(dtBYdtoh,'Initial rate of cooling in degree C/sec');\n", "\n", "// Part(c)\n", "A=4*%pi*r^2;\n", "toh=60;\n", "q_in= h*A*(t_i-t_infinite)*%e^(-h*toh/(rho*l_s*C));// in watt\n", "disp(q_in,'Instantaneous heat transfer rate in watt');\n", "\n", "// Part(d) Total energy transferred during first one minute\n", "V=4/3*%pi*r^3;\n", "TotalEnergy = rho*C*V*(t_i-t_infinite)*(1-%e^(-h*toh/(rho*C*l_s)));\n", "disp(TotalEnergy,'Total energy transferred during first one minute in watt')\n", "\n", "// Note: Answer of first and last part in the book is wrong" ] } , { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 4.3: Time_constant_and_temp_attained_by_junction.sce" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [ "//Exa 4.3\n", "clc;\n", "clear;\n", "close;\n", "//given data\n", "k=40;// in W/mK\n", "rho=8200;// in kg/m^3\n", "C=400;// in J/kgK\n", "D=6*10^-3;// in m\n", "R=D/2;\n", "t_i=30;// in degree C\n", "t_infinite1=400;// for 10 sec in degree C\n", "t_infinite2=20;// for 10 sec in degree C\n", "h=50;// in W/m^2K\n", "\n", "// Part(a)\n", "//l_s= V/A = R/3\n", "l_s= R/3;// in m\n", "//toh= rho*V*C/(h*A) = rho*C*l_s/h\n", "toh= rho*C*l_s/h;// in sec\n", "disp(toh,'Time constance in sec')\n", "\n", "// Part (b)\n", "Bi= h*l_s/k;\n", "// since Bi < 0.1 , hence lumped heat capacity analysis is valid. Now , temperature attained by junction in 10 seconds when exposed to hot air at 400 degree C\n", "toh=10;// in sec\n", "// (t-t_infinite1)/(t_i-t_infinite1)= %e^(-h*A*toh/(rho*V*C)) = %e^(-h*toh/(rho*l_s*C))\n", "t= %e^(-h*toh/(rho*l_s*C))*(t_i-t_infinite1)+t_infinite1;// in degree C\n", "\n", "disp('The junction is taken out from hot air stream and placed in stream of still air 20 degree C. The initial temperature in this case will be '+string(t)+' .')\n", "t_i=t;\n", "toh=20;// in sec\n", "t= %e^(-h*toh/(rho*l_s*C))*(t_i-t_infinite2)+t_infinite2;// in degree C\n", "disp(t,'The temperature attained by junction in degree C');\n", "\n", "// Note: In the last, calculation to find the value of t is wrong so Answer in the book is wrong\n", "\n", "\n", "\n", "\n", "\n", "" ] } , { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 4.4: Time_constant_and_time_required_to_the_temp_change.sce" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [ "//Exa 4.4\n", "clc;\n", "clear;\n", "close;\n", "//given data\n", "k=8;// in W/mK\n", "alpha=4*10^-6;// in m^2/s\n", "h=50;// in W/m^2K\n", "D=6*10^-3;// in m\n", "R=D/2;\n", "T=0.5;// where T = (t-t_infinite)/(t_i-t_infinite)\n", "//l_s= V/A = R/3\n", "l_s= R/2;// in m\n", "Bi= h*l_s/k;\n", "// since Bi < 0.1 , hence lumped heat capacity analysis can be applied\n", "// toh= rho*V*C/(h*A) = rho*C*l_s/h = k*l_s/(h*alpha)\n", "toh= k*l_s/(h*alpha);// in seconds\n", "disp(toh,'time constant in seconds');\n", "// It is given that (t-t_infinite)/(t_i-t_infinite) = 0.5 = %e^(-h*A*c /(rho*V*C)) = %e^(-h*c/(rho*l_s*C)) = %e^(-h*alpha*c/(l_s))\n", "// or (t-t_infinite)/(t_i-t_infinite) = %e^(-h*alpha*c/(l_s);\n", "c= -log(T)*l_s/(h*alpha);// in sec\n", "disp(c,'The time required to temperature change to reach half of its initial value in seconds');\n", "" ] } , { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 4.5: Rate_of_heat_energy_stored.sce" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [ "//Exa 4.5\n", "clc;\n", "clear;\n", "close;\n", "//given data\n", "//t=450-500*x+100*x^2+150*x^3 at any instant, so\n", "// dtBYdx= -500+200*x+450*x^2\n", "\n", "L=0.5;// thickness of the wall in meter\n", "k=10;// in W/mK\n", "// Rate of heating entering in the wall per unit area, at\n", "x=0;\n", "//q1= -k*dtBYdx\n", "q1= -k*(-500+200*x+450*x^2);// in W/m^2\n", "// Rate of heat going out of the wall per unit area , at\n", "x=L;\n", "q2= -k*(-500+200*x+450*x^2);// in W/m^2\n", "E=q1-q2;// in W/m^2\n", "disp(E,'Heat energy stored per unit area in W/m^2')" ] } , { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 4.6: Time_constant_and_time_required_for_the_plate_to_reach_the_temp_of_40_deg_C.sce" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [ "//Exa 4.6\n", "clc;\n", "clear;\n", "close;\n", "//given data\n", "k=385;// in W/mK\n", "h=100;// in W/m^2K\n", "delta =2*10^-3;// thickness of plate in meter\n", "A=25*25;// area of plate in square meter\n", "rho=8800;// kg/m^3\n", "C=400;// J/kg-K\n", "// l_s= V/A= L*B*delta/(2*L*B) = delta/2\n", "l_s= delta/2;// in meter\n", "Bi= h*l_s/k;\n", "// since Bi < 0.1 , hence lumped heat capacity analysis can be applied\n", "\n", "// Part(i)\n", "// toh= rho*V*C/(h*A) = rho*C*l_s/h\n", "toh= rho*C*l_s/h;// in second\n", "disp(toh,'Time constant in seconds');\n", "\n", "// Part(ii)\n", "t_i=400;// in degree C\n", "t=40;// in degree C\n", "t_infinite=25;// in degree C\n", "// (t-t_infinite)/(t_i-t_infinite) = %e^(-h*A*toh /(rho*V*C)) = %e^(-h*toh/(rho*l_s*C)) \n", "toh= -log((t-t_infinite)/(t_i-t_infinite))*rho*C*l_s/h;// in sec\n", "disp(toh,'The time required for the plate to reach the temperature of 40 degree C in seconds');\n", "" ] } , { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 4.7: Time_required_to_cool_plate_to_80_deg_C_and_in_air.sce" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [ "//Exa 4.7\n", "clc;\n", "clear;\n", "close;\n", "//given data\n", "k=380;// in W/mK\n", "delta =6*10^-2;// thickness of plate in meter\n", "rho=8800;// kg/m^3\n", "C=400;// J/kg-K\n", "// l_s= V/A = delta/2\n", "l_s= delta/2;// in meter\n", "t=80;// in degree C\n", "t_i=200;// in degree C\n", "t_inf=30;// in degree C\n", "hw= 75;// in W/m^2K\n", "ha= 10;// in W/m^2K\n", "\n", "// Part(i)\n", "// ha*A*(t-t_inf_a)+ hw*A*(t-t_inf_w) = -rho*V*C*dtBYdtho, since t_ini_a = t_inf_w = t_inf = 30 degree C\n", "// (ha+hw)*A*(t-t_inf)= -rho*V*C*dtBYdtho\n", "// (ha+hw)/(rho*C*V)*A*dtoh = -dt/(t-t_inf)\n", "// integrate('(ha+hw)/(rho*V*C)*A','toh',0,toh) = integrate('1/(t-t_inf)','t',t_i,t)\n", "toh= -rho*l_s*C/(ha+hw)*log((t-t_inf)/(t_i-t_inf));\n", "disp('Time required to cool plate to 80 degree C is : '+string(toh)+' seconds = '+string(toh/60)+' minutes');\n", "\n", "// Part (ii)\n", "t= -rho*l_s*C/(2*ha)*log((t-t_inf)/(t_i-t_inf));\n", "disp('Time required to cool plate in only air is : '+string(t)+' seconds = '+string(t/60)+' minutes');\n", "" ] } , { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 4.8: Maximum_speed_of_ingot_passing_through_the_furnace.sce" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [ "//Exa 4.8\n", "clc;\n", "clear;\n", "close;\n", "//given data\n", "k=45;// in W/m degree C\n", "d =0.1;// in meter\n", "l =0.30;// in meter\n", "t=800;// in degree C\n", "t_i=100;// in degree C\n", "t_infinite=1200;// in degree C\n", "h= 120;// in W/m^2 degree C\n", "alpha=1.2*10^-5;// in meter\n", "rhoC= k/alpha;\n", "V=%pi/4*d^2*l;// in m^3\n", "A= %pi*d*l + 2*%pi/4*d^2;// in m^2\n", "// l_s= V/A = (%pi/4*d^2*l)/(%pi*d*l + 2*%pi/4*d^2) = d*l/(4*l+2*d^2)\n", "l_s = d*l/(4*l+2*d^2);\n", "Bi= h*l_s/k;\n", "// since Bi < 0.1 , hence lumped heat capacity analysis can be applied\n", "// (t-t_infinite)/(t_i-t_infinite) = %e^(-h*A*toh /(rho*V*C)) = %e^(-h*toh/(rho*l_s*C)) = %e^(-h*toh/(rhoC*l_s))\n", "toh = -log((t-t_infinite)/(t_i-t_infinite))*rhoC*l_s/h;// in sec\n", "\n", "// So, the velocity of ingot passing through the furnace\n", "FurnaceLength = 8*100;// in cm\n", "time = toh;\n", "Velocity = FurnaceLength/time;// in cm/sec\n", "disp(Velocity,'Maximum speed in cm/sec')" ] } , { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 4.9: Junction_diamete_and_time_required_for_the_thermocouple_junction.sce" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [ "//Exa 4.9\n", "clc;\n", "clear;\n", "close;\n", "//given data\n", "rho=8500;// in kg/m^3\n", "C=400;// J/kgK\n", "toh=1;// in sec\n", "h= 400;// in W/m^2 degree C\n", "t=198;// in degree C\n", "t_i=25;// in degree C\n", "t_infinite=200;// in degree C\n", "\n", "// Part (1)\n", "// toh =rho*V*C/(h*A) = rho*C*l_s/h\n", "l_s= toh*h/(rho*C);\n", "// l_s = V/A = r/3 \n", "r=3*l_s;// in m\n", "r=r*10^3;// in mm\n", "d=2*r;// in m\n", "disp(d,'Junction diameter needed for the thermocouple in mili miter');\n", "\n", "// Part(ii)\n", "// toh= -rho*V*C/(h*A)*log((t-t_infinite)/(t_i-t_infinite)) \n", "toh = -toh*log((t-t_infinite)/(t_i-t_infinite));\n", "disp(toh,'Time required for the thermocouple junction to reach 198 degree C in seconds');" ] } ], "metadata": { "kernelspec": { "display_name": "Scilab", "language": "scilab", "name": "scilab" }, "language_info": { "file_extension": ".sce", "help_links": [ { "text": "MetaKernel Magics", "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md" } ], "mimetype": "text/x-octave", "name": "scilab", "version": "0.7.1" } }, "nbformat": 4, "nbformat_minor": 0 }