path: root/Introduction_to_Heat_Transfer_by_S._K._Som/Chapter3.ipynb

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   "source": [
    "# Chapter 03:Multidimensional steady-state heat conduction"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Ex3.1:pg-92"
   ]
  },
  {
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   "execution_count": 1,
   "metadata": {
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    {
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     "text": [
      "Introduction to heat transfer by S.K.Som, Chapter 3, Example 1\n",
      "Temperature at the centre in Degree C is\n",
      "T= 125.371641666\n"
     ]
    }
   ],
   "source": [
    "import math\n",
    " \n",
    "print\"Introduction to heat transfer by S.K.Som, Chapter 3, Example 1\"\n",
    "#Length and breadth is given as 1 unit (Gemoetry is Square)\n",
    "L = 1;#length\n",
    "#Problem can be divided into two modules\n",
    "#Solution to module 1 is given by Eq. 3.21, considering the first three terms\n",
    "#n is the looping parameter\n",
    "#theta is the non dimensional temperature defined as ((T-100)/100) where T is actual temperature in degree Celcius.\n",
    "#Initialising theta as zero\n",
    "theta = 0;\n",
    "for n in range(1,3):\n",
    "  theta = theta+((2/math.pi)*((math.sin((n*math.pi)/2)*math.sinh((n*math.pi)/2))*((-1)**(n+1)+1)))/(n*math.sinh(n*math.pi));\n",
    " \n",
    "#Solution to module 2 is given by Eq. 3.24, considering the first three terms\n",
    "for n in range(1,3):\n",
    "  theta2 = theta+(((3*2)/math.pi)*((math.sin((n*math.pi)/2)*math.sinh((n*math.pi)/2))*((-1)**(n+1)+1)))/(n*math.sinh(n*math.pi));\n",
    " \n",
    "#Calculating value of temperature from the value of theta\n",
    "#Temperature in degree celcius\n",
    "print\"Temperature at the centre in Degree C is\"\n",
    "T = theta*100+100\n",
    "print\"T=\",T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Ex3.2:pg-94"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " Introduction to heat transfer by S.K.Som, Chapter 3, Example 2\n",
      "Steady state non dimensional temperature is\n",
      "theta=2*math.sinh(pi*y/a)*math.sin(pi*x/a)/(math.sinh(pi)) + math.sinh(pi*x/a)*math.sin(pi*y/a)/(math.sinh(pi))\n",
      "theta= 0.597805223008\n",
      "Temperature in K at centre point\n",
      "T= 359.780522301\n"
     ]
    }
   ],
   "source": [
    "import math\n",
    " \n",
    "print\"Introduction to heat transfer by S.K.Som, Chapter 3, Example 2\"\n",
    "#Temperature in K at four edges are given\n",
    "#Theta is non dimensional temperature defined as ((T-300)/100) where T is actual temperature in K.\n",
    "#Given length as well as the breadth of square plate is ''a''\n",
    "#Problem can be divided into two modules\n",
    "#Solution to module 1 is given by Eq. 3.23\n",
    "#Solution of first module is non dimensional temperature theta1\n",
    "#theta1=2*math.sinh(pi*y/a)*math.sin(pi*x/a)/(math.sinh(pi))\n",
    "#Solution to module 2 is given by Eq. 3.24\n",
    "#Solution of second module is non dimensional temperature theta2\n",
    "#theta2=math.sinh(pi*x/a)*math.sin(pi*y/a)/(math.sinh(pi))\n",
    "#Therefore\n",
    "print\"Steady state non dimensional temperature is\"\n",
    "print\"theta=2*math.sinh(pi*y/a)*math.sin(pi*x/a)/(math.sinh(pi)) + math.sinh(pi*x/a)*math.sin(pi*y/a)/(math.sinh(pi))\"\n",
    "#At the centre, x coordinate and y coordinate in unit are\n",
    "#x=a/2, y=a/2\n",
    "#Non dimensional temperature at centre point\n",
    "theta = (2*math.sinh(math.pi/2))/math.sinh(math.pi)+math.sinh(math.pi/2)/math.sinh(math.pi);\n",
    "#Temperature in K at centre point\n",
    "print\"theta=\",theta\n",
    "print\"Temperature in K at centre point\"\n",
    "T = theta*100+300\n",
    "print\"T=\",T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Ex3.3:pg-96"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
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     "text": [
      "Introduction to heat transfer by S.K.Som, Chapter 3, Example 3\n",
      "Temperatures at nodal points in degree K\n",
      "T1 in degree K\n",
      "[ 398.67699539  155.83601706   66.53320567  119.43598224   79.06693359\n",
      "   40.99573505   14.15266777    9.19140047]\n",
      "T2 in degree K\n",
      "[  77.91800853  232.60510053   92.0706763    39.53346679   80.21585865\n",
      "   48.72486726   19.11393507   11.30646706]\n",
      "T3 in degree K\n",
      "[  33.26660284   92.0706763   237.56636783   20.49786753   48.72486726\n",
      "   82.33092523   46.76647228   21.51623292]\n",
      "T4 in degree K\n",
      "[  14.15266777   38.22787014   93.53294456    9.19140047   22.61293411\n",
      "   43.03246584  124.91948821   27.99199234]\n",
      "T5 in degree K\n",
      "[-0. -0. -0. -0. -0. -0. -0. -0.]\n",
      "T6 in degree K\n",
      "[-0. -0. -0. -0. -0. -0. -0. -0.]\n",
      "T7 in degree K\n",
      "[  95.65671512  227.3827139   384.21098442   77.62207329  214.83157803\n",
      "  554.32152494  100.40908695  109.12176865]\n",
      "T8 in degree K\n",
      "[  24.51040125   60.30115763  114.75324223   18.87022369   50.97049352\n",
      "  124.71059274   74.64531291  166.55931761]\n"
     ]
    }
   ],
   "source": [
    "import math\n",
    "import numpy\n",
    " \n",
    "print\"Introduction to heat transfer by S.K.Som, Chapter 3, Example 3\"\n",
    "#internodal distance in x direction in m\n",
    "deltax = 1.0/4;\n",
    "#internodal distance in y direction in m\n",
    "deltay = 1.0/4;\n",
    "#Air temperature in degree K\n",
    "Tinfinity = 400;\n",
    "#Heat transfer coefficient in W/(m**2*K)\n",
    "h = 10;\n",
    "#T1, T2, T3, T4, T5, T6, T7, T8 are nodal temperatures in degree K.\n",
    "#T is the temperature matrix and is transpose of [T1 T2 T3 T4 T5 T6 T7 T8]\n",
    "#using Nodal Equations, we have Coefficeint Matrix A as\n",
    "A = [[-4,1,0,0,1,0,0,0],[1,-4,1,0,0,1,0,0],[0,1,-4,1,0,0,1,0],[2,0,0,0,-4,1,0,0],[0,2,0,0,1,-4,1,0],[0,0,2,0,0,1,-4,1],[0,0,2,-6,0,0,0,1],[0,0,0,2,0,0,2,-6]]#Coefficient matrix B\n",
    "B = [[-1200],[-600],[-600],[-600],[0],[0],[-1400],[-800]]\n",
    "\n",
    "\n",
    "#Therefore the temperature matrix is\n",
    "T = numpy.linalg.inv(A)*B;\n",
    "#Temperature at nodal points in degree K\n",
    "print\"Temperatures at nodal points in degree K\"\n",
    "print\"T1 in degree K\"\n",
    "T1 = T[0]\n",
    "print T1\n",
    "print\"T2 in degree K\"\n",
    "T2 = T[1]\n",
    "print T2\n",
    "print\"T3 in degree K\"\n",
    "T3 = T[2]\n",
    "print T3\n",
    "print\"T4 in degree K\"\n",
    "T4 = T[3]\n",
    "print T4\n",
    "print\"T5 in degree K\"\n",
    "T5 = T[4]\n",
    "print T5\n",
    "print\"T6 in degree K\"\n",
    "T6 = T[5]\n",
    "print T6\n",
    "print\"T7 in degree K\"\n",
    "T7 = T[6]\n",
    "print T7\n",
    "print\"T8 in degree K\"\n",
    "T8 = T[7]\n",
    "print T8"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Ex3.5:pg-98"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
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   "outputs": [
    {
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     "text": [
      "Introduction to heat transfer by S.K.Som, Chapter 3, Example 5\n",
      "Temperatures at nodal points in degree C\n",
      "T2 in degree C\n",
      "[  1.83976243e-36   4.79441040e-01   3.66134997e-01   3.07581515e-01\n",
      "   5.38080937e-01]\n",
      "T3 in degree C\n",
      "[  1.46972670e-67   1.92742646e+00   1.47191880e+00   1.23652483e+00\n",
      "   1.07886949e+00]\n",
      "T4 in degree C\n",
      "[  4.52446173e-92   1.47191880e+00   3.54032873e+00   2.97414801e+00\n",
      "   1.67320356e+00]\n",
      "T5 in degree C\n",
      "[  6.50142301e-108   1.23652483e+000   2.97414801e+000   5.91733919e+000\n",
      "   2.36010173e+000]\n",
      "T6 in degree C\n",
      "[  2.06473580e-113   1.16395938e+000   2.79961015e+000   5.57008016e+000\n",
      "   3.18199172e+000]\n"
     ]
    }
   ],
   "source": [
    "import math\n",
    "import numpy\n",
    " \n",
    "print\"Introduction to heat transfer by S.K.Som, Chapter 3, Example 5\"\n",
    "#Thermal conductivity of aluminium in W/(m*K)\n",
    "k = 200.0\n",
    "#Diameter in m\n",
    "d = 20*(10**(-3));\n",
    "#Length of fin in m\n",
    "L = 0.2;\n",
    "#Wall temperature in degree C\n",
    "Tw = 400.0;\n",
    "#Air temperature in degree C\n",
    "Tinfinity = 30;\n",
    "#Heat transfer coefficient in W/(m**2*K)\n",
    "h = 40.0;\n",
    "#internodal distance in x direction in m\n",
    "deltax = L/5;\n",
    "#Node 1 temperature is equal to  wall temperature in degree C\n",
    "T1 = Tw;\n",
    "#using Nodal Equations, we have Coefficeint Matrix A as\n",
    "A = [[2.064,-1,0,0,0],[-1,2.064,-1,0,0],[0,-1,2.064,-1,0],[0,0,-1,2.064,-1],[0,0,0,-1,1.032]]\n",
    "#Coefficient matrix B\n",
    "B = [401.92,1.92,1.92,1.92,0.96]\n",
    "#T2, T3, T4, T5, T6 are nodal temperature in degree C\n",
    "#T is the temperature matrix and is transpose of [T2 T3 T4 T5 T6]\n",
    "#Therefore the temperature matrix is\n",
    "T = numpy.linalg.inv(A)**B;\n",
    "#Temperature at nodal points in degree C\n",
    "print\"Temperatures at nodal points in degree C\"\n",
    "print\"T2 in degree C\"\n",
    "T2 = T[0]\n",
    "print T2\n",
    "print\"T3 in degree C\"\n",
    "T3 = T[1]\n",
    "print T3\n",
    "print\"T4 in degree C\"\n",
    "T4 = T[2]\n",
    "print T4\n",
    "print\"T5 in degree C\"\n",
    "T5 = T[3]\n",
    "print T5\n",
    "print\"T6 in degree C\"\n",
    "T6 = T[4]\n",
    "print T6"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Ex3.6:pg-104"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
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   "outputs": [
    {
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     "output_type": "stream",
     "text": [
      "Introduction to heat transfer by S.K.Som, Chapter 3, Example 6\n",
      "Temperatures at nodal points in degree C\n",
      "T1 in degree C\n",
      "[  -0.           -0.           -0.          186.04651163    1.86046512\n",
      "    2.79069767    1.86046512    0.46511628   74.41860465]\n",
      "T2 in degree C\n",
      "[  -0.           -0.           -0.           74.41860465    1.69263965\n",
      "    3.56988732    2.51738192    0.62934548  157.39630784]\n",
      "T3 in degree C\n",
      "[ -0.          -0.          -0.          83.72093023   3.88875569\n",
      "   4.80220571   3.06401343   0.76600336  65.85950611]\n",
      "T4 in degree C\n",
      "[ -0.          -0.          -0.          55.81395349   2.45504675\n",
      "   5.7444258    4.10453129   1.02613282  77.58331335]\n",
      "T5 in degree C\n",
      "[ -0.          -0.          -0.          37.20930233   1.77415488\n",
      "   4.6199952    7.34116519   1.8352913   51.37856629]\n",
      "T6 in degree C\n",
      "[ -0.          -0.          -0.          37.20930233   8.78446416\n",
      "   4.92927356   2.18652601   0.5466315   33.8527931 ]\n",
      "T7 in degree C\n",
      "[ -0.          -0.          -0.          27.90697674   2.46463678\n",
      "   9.98561496   3.49556461   0.87389115  35.69887317]\n",
      "T8 in degree C\n",
      "[ -0.          -0.          -0.          18.60465116   1.09326301\n",
      "   3.49556461  10.57779909   2.64444977  25.17381923]\n",
      "T9 in degree C\n",
      "[ -0.          -0.          -0.           9.30232558   0.5466315\n",
      "   1.74778231   5.28889954  11.32222489  12.58690961]\n"
     ]
    }
   ],
   "source": [
    "import math\n",
    " \n",
    "print\"Introduction to heat transfer by S.K.Som, Chapter 3, Example 6\"\n",
    "#Thermal conductivity of concrete in W/mK\n",
    "k = 2;\n",
    "#Length in m\n",
    "L = 0.2;\n",
    "#Breadth in m\n",
    "b = 0.2;\n",
    "#Depth in m\n",
    "d = 0.2;\n",
    "#Temperature of hot gas in chimney in degree C\n",
    "Tg = 400;\n",
    "#Air temperature in degree C\n",
    "Tinfinity = 20;\n",
    "#internodal distance in x direction in m\n",
    "deltax = 0.1;\n",
    "#internodal distance in y direction in m\n",
    "deltay = 0.1;\n",
    "#Heat transfer coefficient in W/(m**2*K)\n",
    "h = 20;\n",
    "#T1, T2, T3, T4, T5, T6, T7, T8, T9 are nodal temperatures in degree K.\n",
    "#T is the temperature matrix and is transpose of [T1 T2 T3 T4 T5 T6 T7 T8 T9]\n",
    "#using Nodal Equations, we have Coefficeint Matrix A as\n",
    "A = numpy.array([[1,0,-4,2,0,1,0,0,0],[0,1,1,-4,1,0,1,0,0],[0,0,0,2,-4,0,0,2,0],[-3,1,1,0,0,0,0,0,0],[0,0,1,0,0,-3,1,0,0],[0,0,0,2,0,1,-6,1,0],[0,0,0,0,2,0,1,-6,1],[0,0,0,0,0,0,0,1,-2],[1,-4,0,2,0,0,0,0,0]]);\n",
    "#Coefficient matrix B\n",
    "B = numpy.array([0,0,0,-400,-20,-40,-40,-20,-400]);\n",
    "#Therefore the temperature matrix is\n",
    "T = numpy.linalg.inv(A)*B;\n",
    "#Temperature at nodal points in degree C\n",
    "print\"Temperatures at nodal points in degree C\"\n",
    "print\"T1 in degree C\"\n",
    "T1 = T[0]\n",
    "print T1\n",
    "print\"T2 in degree C\"\n",
    "T2 = T[1]\n",
    "print T2\n",
    "print\"T3 in degree C\"\n",
    "T3 = T[2]\n",
    "print T3\n",
    "print\"T4 in degree C\"\n",
    "T4 = T[3]\n",
    "print T4\n",
    "print\"T5 in degree C\"\n",
    "T5 = T[4]\n",
    "print T5\n",
    "print\"T6 in degree C\"\n",
    "T6 = T[5]\n",
    "print T6\n",
    "print\"T7 in degree C\"\n",
    "T7 = T[6]\n",
    "print T7\n",
    "print\"T8 in degree C\"\n",
    "T8 = T[7]\n",
    "print T8\n",
    "print\"T9 in degree C\"\n",
    "T9 = T[8]\n",
    "print T9"
   ]
  }
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