path: root/Fundamentals_of_Fluid_Mechanics/Ch_7.ipynb

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   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Chapter 7 :Similitude ,Dimensional Analysis and Modelling"
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Example 7.5 Page no.372"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "\n",
      "D=0.1                            #m\n",
      "H=0.3                            #m\n",
      "v=50.0                           #km/hr\n",
      "Dm=20.0                          #mm\n",
      "T=20.0                           #degree C\n",
      "fm=49.9                          #Hz  frequency for the model\n",
      "#f=func(D,H,V,d,vis)\n",
      "#f=T**(-1) D=l H=L V=L*(T**(-1)) d=M*(L**(-3)) vis=M*(L**(-1))*(T**(-1))\n",
      "#by applying pi theorem,\n",
      "#(f*D/V)=funct((D/H),(d*V*D/vis))\n",
      "#hence Dm/Hm = D/H, dm*Vm*Dm/vism = d*V*D/vis, and (f*D/V)=(fm*Dm/Vm)\n",
      "\n",
      "#calculation\n",
      "Hm=(Dm*H*1000/(D*1000))   #mm\n",
      "V=v*1000/3600             #m/s\n",
      "vism=0.001                #kg/(m*s)\n",
      "vis=1.79*10**-5           #kg/(m*s)\n",
      "d=1.23                        #kg/(m**3)\n",
      "dm=998                        #kg/(m**3)\n",
      "Vm=(vism*d*D*V*1000)/(vis*dm*Dm)        #m/s\n",
      "f=(V/Vm)*(Dm/(D*1000))*fm               #Hz\n",
      "\n",
      "#result\n",
      "print \"The model dimension =\",round(Hm,3),\"mm\"\n",
      "print \"The velocity at which the test should be performed=\",round(Vm,2),\"m/s\"\n",
      "print \"The predicted prototype vortex shredding frequency =\",round(f,1),\"Hz\"\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The model dimension = 60.0 mm\n",
        "The velocity at which the test should be performed= 4.781 m/s\n",
        "The predicted prototype vortex shredding frequency = 29.0 Hz\n"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Example 7.6 Page no.377"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%pylab inline"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "Welcome to pylab, a matplotlib-based Python environment [backend: module://IPython.zmq.pylab.backend_inline].\n",
        "For more information, type 'help(pylab)'.\n"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "\n",
      "D=2.0                              #ft\n",
      "Q=30.0                           #cfs\n",
      "Dm=3.0                          #in\n",
      "#Rem=Re hence (Vm*Dm/kvism)=(V*D/kvis) where kvis is kinematic viscosity\n",
      "#kvis=kvism same fluid is used for model and prototype\n",
      "#(Vm/V)=(D/Dm)\n",
      "#Q=VA hence Qm/Q = (Vm*Am)/(V*A)=(Dm/D)\n",
      "\n",
      "#  calculation\n",
      "Qm=(Dm/12)*Q/D          #cfs\n",
      "\n",
      "#result\n",
      "print \"The required flowrate in the model=\",Qm,\"cfs\"\n",
      "\n",
      "#Plot\n",
      "import matplotlib.pyplot as plt\n",
      "fig = plt.figure()\n",
      "ax = fig.add_subplot(111)\n",
      "ax.plot([0.125], [8], 'o')\n",
      "ax.annotate('(0.125,8)', xy=(0.125,8.5))\n",
      "\n",
      "T=[0.04,0.125,0.2,0.6,1]\n",
      "K=[25,8,5,2,1]\n",
      "xlabel(\"Dm/D\") \n",
      "ylabel(\"Vm/V\") \n",
      "plt.xlim((0,1))\n",
      "plt.ylim((0,25))\n",
      "a=plot(T,K)\n",
      "show(a)\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The required flowrate in the model= 3.75 cfs\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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7jR22b9+OqVOnorq6Gm+++SZOnjwJAIiIiMD06dPRp0+fVp9/+OGHkZaWhg0bNrR6vmfP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\nTkSuQmCC\n"
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Example 7.7 Page no.380"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "\n",
      "V=240.0                             #mph\n",
      "ratio=0.1\n",
      "Vair=240.0                          #mph\n",
      "Fm=1.0                               #lb Fm =drag force on model\n",
      "p=14.7                              #psia standard atmospheric pressure\n",
      "#Re=Rem\n",
      "#(d*V*l/vis)=(dm*Vm*lm/vism)\n",
      "#here Vm=V and lm/l=ratio\n",
      "#assumption made is that an increase in pressure does not significantly change viscosity\n",
      "\n",
      "#calculation\n",
      "drat=V/(ratio*Vair)             #where drat=dm/d\n",
      "\n",
      "#for an ideal gas p=d*R*T\n",
      "#T=Tm\n",
      "#hence, pm/p=dm/d pm/p=prat\n",
      "pm=p*drat\n",
      "#F/(0.5*d*(V**2)*(l**2))=Fm/(0.5*dm*(Vm**2)*(lm**2))\n",
      "F=(1/drat)*((V/Vair)**2)*((1/ratio)**2)*Fm\n",
      "\n",
      "#result\n",
      "print \"The required air pressure in the tunnel=\",round(pm,3),\"psia\"\n",
      "print \"The corrosponding drag on the prtotype \\nfor a 1 lb drag on the model=\",round(F,3),\"lb\""
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The required air pressure in the tunnel= 147.0 psia\n",
        "The corrosponding drag on the prtotype \n",
        "for a 1 lb drag on the model= 10.0 lb\n"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Example 7.8 Page no.384"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "\n",
      "w=20.0                           #m\n",
      "Q=125.0                        #(m**3)/s\n",
      "ratio=0.066\n",
      "t=24.0                           #hours\n",
      "wm=ratio*w              #m\n",
      "#Vm/(gm*lm)**0.5 = V/(g*l)**0.5\n",
      "#gm=g\n",
      "#Q=VA and lm/l=1/15\n",
      "#hence Qm/Q = ((lm/l)**0.5)*((lm/l)**2) = ratio**2.5\n",
      "Qm=(ratio**2.5)*Q\n",
      "#V=l/t\n",
      "#tm/t=(V/Vm)*(lm/l)=ratio**0.5\n",
      "tm=(ratio**0.5)*t         #hours\n",
      "\n",
      "#result\n",
      "print \"The required model width=\",round(wm,3),\"m\" \n",
      "print \"The required model flowrate=\",round(Qm,3),\"m**3/s\"\n",
      "print \"The operating time for the model=\",round(tm,1),\"hrs\"\n",
      "\n",
      "#plot\n",
      "import matplotlib.pyplot as plt\n",
      "fig = plt.figure()\n",
      "ax = fig.add_subplot(111)\n",
      "\n",
      "T=[0.02,0.07,0.2,0.4,0.5]\n",
      "K=[4,6.20,10,15,17]\n",
      "xlabel(\"lm/l\") \n",
      "ylabel(\"tm  (hr)\") \n",
      "plt.xlim((0,0.5))\n",
      "plt.ylim((0,20))\n",
      "\n",
      "ax.plot([0.07], [6.20], 'o')\n",
      "ax.annotate('(1/15,6.20hr)', xy=(0.1,6))\n",
      "a=plot(T,K)\n",
      "show(a)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The required model width= 1.32 m\n",
        "The required model flowrate= 0.14 m**3/s\n",
        "The operating time for the model= 6.2 hrs\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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      }
     ],
     "prompt_number": 3
    }
   ],
   "metadata": {}
  }
 ]
}