Supporting information.ipynb

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    "#  CHE 614 course project final report (supporting information): Heat transfer behavior modeling in core-shell paraffin@Silica nanoparticles with Openpnm.\n",
    "## Main content: shell thickness effect on effective thermal conductance in different temperature reigons\n",
    "## Jinhe Zhang #20985826\n",
    "As discussed at the beginning of the semester, this section supplements the heat transfer behavior of materials in different temperature ranges. The three situations are mainly analyzed, the inlet and outlet temperature is higher than the paraffin phase change temperature, the inlet and outlet temperature is lower than the paraffin phase change temperature, and the paraffin phase change temperature is between the inlet and outlet temperature."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "439ab540",
   "metadata": {},
   "source": [
    "## 1. Generating network"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "eab27704",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import openpnm as op\n",
    "import scipy as sp\n",
    "%config InlineBackend.figure_formats = ['svg']\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "0d736ad5",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "------------------------------------------------------------\n",
      "CRITICAL   : front and back labels have been switched to obey the right-hand rule \n",
      "SOURCE     : openpnm.network.Cubic.__init__ \n",
      "TIME STAMP : 2022-08-25 19:55:09,358\n",
      "------------------------------------------------------------\n"
     ]
    }
   ],
   "source": [
    "shape = [10, 10, 10]\n",
    "spacing = 1e-5\n",
    "net = op.network.Cubic(shape=shape, spacing=spacing)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "debed2b2",
   "metadata": {},
   "source": [
    "## 2. Adding geometry"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "1e229c24",
   "metadata": {},
   "outputs": [],
   "source": [
    "geom = op.geometry.GenericGeometry(network=net, pores=net.Ps, throats=net.Ts)\n",
    "geom['pore.diameter'] = np.random.rand(net.Np)*1e-5\n",
    "mode = op.models.misc.from_neighbor_pores\n",
    "geom['throat.diameter'] = mode(target=geom, prop='pore.diameter')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8116d482",
   "metadata": {},
   "source": [
    "## 3. Adding phases"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 169,
   "id": "2fc5f6d4",
   "metadata": {},
   "outputs": [],
   "source": [
    "prf = op.phases.GenericPhase(network=net)\n",
    "air = op.phases.Air(network=net)\n",
    "Ka = air['pore.thermal_conductivity']   # air heat transfer coefficient\n",
    "Kps = 0.4  #solid paraffin heat transfer coefficient\n",
    "Kpl = 0.2  #liquid paraffin heat transfer coefficient\n",
    "Ks = 0.27  #silica heat transfer coefficient"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4156840e",
   "metadata": {},
   "source": [
    "## 4. Adding physics"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 170,
   "id": "684c348b",
   "metadata": {},
   "outputs": [],
   "source": [
    "phys = op.physics.GenericPhysics(network=net, phase=prf, geometry=geom)\n",
    "P12 = net['throat.conns']\n",
    "D1 = geom['pore.diameter'][P12[:,0]]  # diameter of one pore/ paraffin core\n",
    "D2 = geom['pore.diameter'][P12[:,1]]  # diameter of another pore/ paraffin core\n",
    "R_T = geom['throat.diameter']/2  # radius of throat/ heat transfer channel\n",
    "R_P = geom['pore.diameter']/2  # radius of pore/ paraffin core\n",
    "R_w = spacing - D1/2 - D2/2  # the R_w means the distance between two paraffin sphere surfaces\n",
    "Rs = 1.0E-06  # silica layer thickness\n",
    "def gfunc(T, Kp):\n",
    "        g_P = Kp*sp.pi*(R_P)**2/(R_P)\n",
    "        g_th = np.zeros_like(R_T)              \n",
    "        for i in range(len(R_w)):\n",
    "            if R_w[i] - 2 * Rs > 0:  #means the air gap exist between two spheres\n",
    "                R_sas = R_w - 2*Rs             \n",
    "                g_T = np.mean(Ks)*sp.pi*(R_T[i])**2/(Rs)\n",
    "                g_A = np.mean(Ka)*sp.pi*(R_T[i])**2/(R_sas[i])\n",
    "                g_th[i] = (1/g_A + 2/g_T + 1/g_P[P12[i, 0]] + 1/g_P[P12[i, 1]])**-1\n",
    "            else:   #means two spheres are connected\n",
    "                g_T = np.mean(Ks)*sp.pi*(R_T[i])**2/(R_w[i])\n",
    "                g_th[i] = (1/g_T + 1/g_P[P12[i, 0]] + 1/g_P[P12[i, 1]])**-1\n",
    "        return g_th\n",
    "phys['throat.thermal_conductance'] = gfunc(T=293.0, Kp=0.4)  #temporarily assuming all paraffin are in solid state"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a06215db",
   "metadata": {},
   "source": [
    "## 5. Performing fourier conduction\n",
    "### 5.1. T_in and T_out are higher than phase chage temperature "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 171,
   "id": "99d6f8d6",
   "metadata": {},
   "outputs": [],
   "source": [
    "inlet = net.pores('left')  # set the inlet\n",
    "outlet = net.pores('right')  # set the outlet\n",
    "T_in = 353  # K, inlet temperature\n",
    "T_out = 333  # K, outlet temperature\n",
    "T_melting = 323  # K, paraffin melting temperature\n",
    "fc = op.algorithms.FourierConduction(network=net)\n",
    "fc.setup(phase=prf)\n",
    "fc.set_value_BC(pores=inlet, values=T_in)\n",
    "fc.set_value_BC(pores=outlet, values=T_out)\n",
    "fc.run()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 172,
   "id": "133c1463",
   "metadata": {},
   "outputs": [],
   "source": [
    "num = 5  # times of iterate\n",
    "for x in range(num):\n",
    "    T_p = fc['pore.temperature']\n",
    "    K_prf = np.zeros_like(R_P)\n",
    "    for i in range(len(T_p)):\n",
    "        if T_p[i] > T_melting:\n",
    "            K_prf[i] = Kpl\n",
    "        else:\n",
    "            K_prf[i] = Kps\n",
    "    prf['pore.thermal_conductivity'] = K_prf\n",
    "    phys['throat.thermal_conductance'] = gfunc(T = T_p, Kp=K_prf)\n",
    "    fc = op.algorithms.FourierConduction(network=net)\n",
    "    fc.setup(phase=prf)\n",
    "    fc.set_value_BC(pores=inlet, values=T_in)\n",
    "    fc.set_value_BC(pores=outlet, values=T_out)\n",
    "    fc.run()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 173,
   "id": "2140f34e",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.243735E-02\n"
     ]
    }
   ],
   "source": [
    "rate_inlet = fc.rate(pores=inlet)[0]\n",
    "A = (shape[0] * shape[1])*(spacing**2)\n",
    "L = shape[2]*spacing\n",
    "k_eff = rate_inlet * L / (A * (T_in - T_out))\n",
    "print(\"{0:.6E}\".format(k_eff))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "1d0b3731",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "------------------------------------------------------------\n",
      "WARNING    : algorithm | alg_06 | properties | pore.bc_rate has nans, will not write to file \n",
      "SOURCE     : openpnm.io.VTK.export_data \n",
      "TIME STAMP : 2022-08-25 20:04:26,300\n",
      "------------------------------------------------------------\n",
      "------------------------------------------------------------\n",
      "WARNING    : algorithm | alg_06 | properties | pore.bc_value has nans, will not write to file \n",
      "SOURCE     : openpnm.io.VTK.export_data \n",
      "TIME STAMP : 2022-08-25 20:04:26,301\n",
      "------------------------------------------------------------\n"
     ]
    }
   ],
   "source": [
    "net.project.export_data(phases=[fc], filename='80to60_ls_prf', filetype='vtk')"
   ]
  },
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   "cell_type": "markdown",
   "id": "26b04eb4",
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  {
   "cell_type": "code",
   "execution_count": 81,
   "id": "32a7ce3c",
   "metadata": {},
   "outputs": [
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    "plt.plot(x,y,color='r',marker='o',linestyle='dashed')\n",
    "plt.xlabel('Shell thickness')\n",
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    "            arrowprops=dict(facecolor='grey', shrink=0.01),\n",
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    "plt.show()"
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  {
   "cell_type": "markdown",
   "id": "73a879c1",
   "metadata": {},
   "source": [
    "### 5.2. T_in and T_out are lower than phase chage temperature "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 174,
   "id": "ae287e61",
   "metadata": {},
   "outputs": [],
   "source": [
    "inlet = net.pores('left')  # set the inlet\n",
    "outlet = net.pores('right')  # set the outlet\n",
    "T_in = 313  # K, inlet temperature\n",
    "T_out = 293  # K, outlet temperature\n",
    "T_melting = 323  # K, paraffin melting temperature\n",
    "fc = op.algorithms.FourierConduction(network=net)\n",
    "fc.setup(phase=prf)\n",
    "fc.set_value_BC(pores=inlet, values=T_in)\n",
    "fc.set_value_BC(pores=outlet, values=T_out)\n",
    "fc.run()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 175,
   "id": "05c5fa1a",
   "metadata": {},
   "outputs": [],
   "source": [
    "num = 5  # times of iterate\n",
    "for x in range(num):\n",
    "    T_p = fc['pore.temperature']\n",
    "    K_prf = np.zeros_like(R_P)\n",
    "    for i in range(len(T_p)):\n",
    "        if T_p[i] > T_melting:\n",
    "            K_prf[i] = Kpl\n",
    "        else:\n",
    "            K_prf[i] = Kps\n",
    "    prf['pore.thermal_conductivity'] = K_prf\n",
    "    phys['throat.thermal_conductance'] = gfunc(T = T_p, Kp=K_prf)\n",
    "    fc = op.algorithms.FourierConduction(network=net)\n",
    "    fc.setup(phase=prf)\n",
    "    fc.set_value_BC(pores=inlet, values=T_in)\n",
    "    fc.set_value_BC(pores=outlet, values=T_out)\n",
    "    fc.run()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 176,
   "id": "ebad534a",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.593873E-02\n"
     ]
    }
   ],
   "source": [
    "rate_inlet = fc.rate(pores=inlet)[0]\n",
    "A = (shape[0] * shape[1])*(spacing**2)\n",
    "L = shape[2]*spacing\n",
    "k_eff = rate_inlet * L / (A * (T_in - T_out))\n",
    "print(\"{0:.6E}\".format(k_eff))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 182,
   "id": "f091396c",
   "metadata": {},
   "outputs": [
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    "y=[7.41E-03, 8.39E-03, 9.67E-03, 1.13E-02, 1.33E-02, 1.59E-02, 1.88E-02, 2.18E-02, 2.49E-02, 2.79E-02, 3.08E-02]\n",
    "plt.plot(x,y,color='b',marker='o',linestyle='dashed')\n",
    "plt.xlabel('Shell thickness')\n",
    "plt.ylabel('K_eff')\n",
    "plt.annotate('Initial condition', xy=(1.00E-06,1.70E-02), xytext=(0.75E-06, 2.60E-02),\n",
    "            arrowprops=dict(facecolor='grey', shrink=0.01),\n",
    "            )\n",
    "plt.show()"
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   "source": [
    "### 5.3. Phase change temperature is between T_in and T_out\n",
    "This part has the same conditions in the final report. However, the network is randomly built, which makes the solution different from the final report."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 177,
   "id": "5af052b9",
   "metadata": {},
   "outputs": [],
   "source": [
    "inlet = net.pores('left')  # set the inlet\n",
    "outlet = net.pores('right')  # set the outlet\n",
    "T_in = 333  # K, inlet temperature\n",
    "T_out = 313  # K, outlet temperature\n",
    "T_melting = 323  # K, paraffin melting temperature\n",
    "fc = op.algorithms.FourierConduction(network=net)\n",
    "fc.setup(phase=prf)\n",
    "fc.set_value_BC(pores=inlet, values=T_in)\n",
    "fc.set_value_BC(pores=outlet, values=T_out)\n",
    "fc.run()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 178,
   "id": "87be6519",
   "metadata": {},
   "outputs": [],
   "source": [
    "num = 5  # times of iterate\n",
    "for x in range(num):\n",
    "    T_p = fc['pore.temperature']\n",
    "    K_prf = np.zeros_like(R_P)\n",
    "    for i in range(len(T_p)):\n",
    "        if T_p[i] > T_melting:\n",
    "            K_prf[i] = Kpl\n",
    "        else:\n",
    "            K_prf[i] = Kps\n",
    "    prf['pore.thermal_conductivity'] = K_prf\n",
    "    phys['throat.thermal_conductance'] = gfunc(T = T_p, Kp=K_prf)\n",
    "    fc = op.algorithms.FourierConduction(network=net)\n",
    "    fc.setup(phase=prf)\n",
    "    fc.set_value_BC(pores=inlet, values=T_in)\n",
    "    fc.set_value_BC(pores=outlet, values=T_out)\n",
    "    fc.run()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 179,
   "id": "8931a8c4",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.409432E-02\n"
     ]
    }
   ],
   "source": [
    "rate_inlet = fc.rate(pores=inlet)[0]\n",
    "A = (shape[0] * shape[1])*(spacing**2)\n",
    "L = shape[2]*spacing\n",
    "k_eff = rate_inlet * L / (A * (T_in - T_out))\n",
    "print(\"{0:.6E}\".format(k_eff))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 187,
   "id": "e41e8ecc",
   "metadata": {},
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       "</svg>\n"
      ],
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
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     "output_type": "display_data"
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   ],
   "source": [
    "x=[0.00E+00, 2.00E-07,4.00E-07,6.00E-07,8.00E-07,1.00E-06,1.20E-06,1.40E-06,1.60E-06,1.80E-06,2.00E-06]\n",
    "y1=[6.68E-03, 7.42E-03, 8.34E-03, 9.48E-03, 1.08E-02, 1.24E-02, 1.42E-02, 1.60E-02, 1.78E-02, 1.97E-02, 2.15E-02]\n",
    "y2=[7.41E-03, 8.39E-03, 9.67E-03, 1.13E-02, 1.33E-02, 1.59E-02, 1.88E-02, 2.18E-02, 2.49E-02, 2.79E-02, 3.08E-02]\n",
    "y3=[7.03E-03, 7.89E-03, 8.98E-03, 1.04E-02, 1.20E-02, 1.41E-02, 1.64E-02, 1.87E-02, 2.11E-02, 2.36E-02, 2.61E-02]\n",
    "plt.plot(x,y1,color='red',marker='o',linestyle='dashed', label=\"T_in, T_out > T_melting\")\n",
    "plt.plot(x,y2,color='blue',marker='o',linestyle='dashed', label=\"T_in, T_out < T_melting\")\n",
    "plt.plot(x,y3,color='green',marker='o',linestyle='dashed', label=\"T_in > T_melting > T_out\")\n",
    "plt.xlabel('Shell thickness')\n",
    "plt.ylabel('K_eff')\n",
    "plt.legend(loc=\"best\", fontsize=10)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "cd0cbd6a",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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