Process_layouts.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 4. Membrane Processes Layouts"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this set of notes we will use an example ultrafiltration process to introduce material balances in different process configurations. We will consider a Batch, semi-Batch (feed and bleed) and continuous membrane processes to develop general ideas around material balances in membrane units. Every example will be introduced by a slight variant of a common problem statement. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4.1 Batch"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Problem Statement__ \n",
    "\n",
    "500 l of fruit juice are concentrated from an initial solid-residue content of 0.05 kg/l to a solid-residue content of 0.2 kg/l through a batch microfiltration process. The total area of the membrane is of 20 $\\mathrm{m^2}$ and the flux of pure water through the membrane is captured by the empirical expression: $J=BC^{-1} [m/h]$ with $B=0.1$ where C is the solid residue concentration in kg/l. Compute the process time necessary to reach the target concentration.\n",
    "\n",
    "<img src=\"./batch.png\" alt=\"Drawing\" style=\"width: 300px;\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Solution trace__\n",
    "\n",
    "The global differential material balance (volume basis) reads: \n",
    "\n",
    "$$\n",
    "\\frac{dV}{dt}=-J\\,A\n",
    "$$(PLYeq1)\n",
    "\n",
    "where $V$ is the volume in the concentrate loop, $J$ the permeate flux per unit area and $A$ the total membrane area. \n",
    "The permeate flux is defined by the expression: \n",
    "\n",
    "$$\n",
    "J=BC^{-1}\n",
    "$$(PLYeq2)\n",
    "\n",
    "with $B=0.1$. \n",
    "\n",
    "Since the solid residue never leaves the retentate loop,  the differential material balance on the solid residue reads: \n",
    "\n",
    "$$\n",
    "\\frac{dm}{dt}=0 \n",
    "$$(PLYeq3)\n",
    "\n",
    "Which means that the total mass of solid residue is constant and equal to its initial mass:\n",
    "\n",
    "$$\n",
    "m=m_0=V_0C_0 \n",
    "$$(PLYeq4)\n",
    "\n",
    "The global differential material balance can thus be rewritten as: \n",
    "\n",
    "$$\n",
    "\\frac{dV}{dt}=-J\\,A=-\\frac{B}{V_0C_0 }AV\n",
    "$$(PLYeq5)\n",
    "\n",
    "In this simple case it can be integrated analytically to compute the volume as a function of time: \n",
    "\n",
    "$$\n",
    "\\int_{V_0}^{V^{\\prime}}\\frac{dV}{V}=\\int_0^{t^{\\prime}}-\\frac{B}{V_0C_0 }Adt\n",
    "$$(PLYeq6)\n",
    "\n",
    "$$\n",
    "\\int_{V_0}^{V^{\\prime}}\\frac{dV}{V}=\\left[\\ln{V}\\right]^{V^{\\prime}}_{V_0}=\\ln\\left(\\frac{V^\\prime}{V_0}\\right)\n",
    "$$(PLYeq7)\n",
    "\n",
    "$$\n",
    "\\int_0^{t^{\\prime}}-\\frac{B}{V_0C_0 }Adt=-\\frac{B}{V_0C_0 }At^\\prime\n",
    "$$(PLYeq8)\n",
    "\n",
    "Putting together both sides of the equation and substituting $V^\\prime$ with $V$ for the sake of simplicity in the notation we get: \n",
    "\n",
    "$$\n",
    "V(t)=V_0e^{-\\frac{B}{V_0C_0 }At}\n",
    "$$(PLYeq9)\n",
    "\n",
    "which describes the time dependence of the volume in the batch membrane separator. \n",
    "\n",
    "The concentration can thus be calculated as: \n",
    "\n",
    "$$\n",
    "C(t)=\\frac{m}{V(t)}=\\frac{C_0V_0}{V_0}e^{\\frac{B}{V_0C_0 }At}={C_0}e^{\\frac{B}{V_0C_0 }At}\n",
    "$$(PLYeq10)\n",
    "\n",
    "The process time necessary to obtain a solid residue concentration of 0.2 $[kg/l]$ is of 17.3 $[h]$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Numerical solution__"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Applications/anaconda3/lib/python3.7/site-packages/numpy/core/shape_base.py:65: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray.\n",
      "  ary = asanyarray(ary)\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The process time necessary to obtain a solid residue concentration of 200.0 [kg/l] is: [17.32867951] [h]\n"
     ]
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt \n",
    "from scipy.optimize import fsolve\n",
    "\n",
    "# Parameters: \n",
    "N = 500 #number of points\n",
    "time = np.linspace(0, 30, N)\n",
    "\n",
    "#Every length in m\n",
    "\n",
    "A=20;  #m^2\n",
    "B=0.1;\n",
    "C0=0.05*1E3; #kg /l * dm^3/m^3\n",
    "V0=500*1E-3; #l * m^3/dm^3\n",
    "\n",
    "C_specific = 0.2*1E3\n",
    "\n",
    "#Operating Equation\n",
    "C = C0*np.exp(B*A/V0/C0*time) \n",
    "V = V0*np.exp(-B*A/V0/C0*time) \n",
    "\n",
    "def equation(proc_time):\n",
    "    eq1 = C0*np.exp(B*A/V0/C0*proc_time) - C_specific\n",
    "    return eq1\n",
    "\n",
    "process_time = fsolve(equation,[1])\n",
    "\n",
    "#Plotting\n",
    "figure=plt.figure()\n",
    "axes = figure.add_axes([0.1,0.1,0.8,0.8])\n",
    "plt.xticks(fontsize=14)\n",
    "plt.yticks(fontsize=14)\n",
    "axes.plot(time,V, marker=' ' , color='b')\n",
    "\n",
    "plt.title('Suspension Volume', fontsize=18);\n",
    "axes.set_xlabel('time [h]', fontsize=14);\n",
    "axes.set_ylabel('V [m$^3$]',fontsize=14);\n",
    "\n",
    "figure=plt.figure()\n",
    "axes = figure.add_axes([0.1,0.1,0.8,0.8])\n",
    "plt.xticks(fontsize=14)\n",
    "plt.yticks(fontsize=14)\n",
    "axes.plot(time,C, marker=' ' , color='r')\n",
    "axes.plot([process_time,process_time],[0, C_specific], marker=' ' , color='lime', markersize=3)\n",
    "axes.plot([0,process_time],[C_specific, C_specific], marker=' ' , color='lime', markersize=3)\n",
    "\n",
    "\n",
    "axes.plot([process_time,process_time],[0, C_specific], marker=' ' , color='lime', markersize=3)\n",
    "\n",
    "\n",
    "axes.plot(process_time,C_specific, marker='o' , color='lime', markersize=10)\n",
    "\n",
    "plt.title('Solute concentration', fontsize=18);\n",
    "axes.set_xlabel('time [h]', fontsize=14);\n",
    "axes.set_ylabel('concentration [kg/m$^3$]',fontsize=14);\n",
    "\n",
    "print(\"The process time necessary to obtain a solid residue concentration of\", C_specific, \"[kg/l] is:\", process_time, \"[h]\") "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4.2 Feed and Bleed"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Feed and Bleed:  Problem Statement__\n",
    "\n",
    "500 l of fruit juice are concentrated in 5 h of steady state operation from an initial solid-residue content of 0.05 $kg/l$ through a feed and bleed process. The total area of the membrane is of 20 $m^2$ and the flux of pure water through the membrane is captured by the empirical expression: $J=BC^{-1} [m/h]$ with $B=0.1$ where C is the solid residue concentration in kg/l. Is the steady state concentration of the retentate compatible with the specifics of 0.2 $kg/l$? \n",
    "\n",
    "<img src=\"feedandbleed.png\" alt=\"Drawing\" style=\"width: 300px;\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Feed and Bleed: Solution trace__\n",
    "\n",
    "Also in this case the global differential material balance and the solid-residue material balance can be written as follows: \n",
    "\n",
    "$$\n",
    "\\frac{dV}{dt}=Q_{IN}-J\\,A-Q_{OUT}=0\n",
    "$$(PLYeq11)\n",
    "\n",
    "$$\n",
    "\\frac{dm}{dt}=Q_{IN}C_{IN}-Q_{OUT}C_{OUT}=0\n",
    "$$(PLYeq12)\n",
    "\n",
    "At steady state ${dV}/{dt}=0$ as well as ${dm}/{dt}=0$. The two ODEs become then two algebraic equations that should be solved together to compute the steady state concentration. \n",
    "From the global material balance we get: \n",
    "\n",
    "$$\n",
    "Q_{OUT}=Q_{IN}-J\\,A=Q_{IN}-\\frac{B}{C_{OUT}}A\n",
    "$$(PLYeq13)\n",
    "\n",
    "and then, with some manipulations: \n",
    "\n",
    "$$\n",
    "C_{OUT}=C_{IN}+\\frac{AB}{Q_{IN}}\n",
    "$$(PLYeq14)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The steady state concentration is 70.0 [kg/m^3]\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt \n",
    "from scipy.optimize import fsolve\n",
    "\n",
    "# data: \n",
    "A=20;  #[m^2]\n",
    "B=0.1;\n",
    "C0=0.05*1E3; #kg /l * dm^3/m^3\n",
    "V0=500*1E-3; #l * m^3/dm^3\n",
    "process_time=5; # [h]\n",
    "\n",
    "Cout=C0+A*B*process_time/V0\n",
    "\n",
    "\n",
    "print(\"The steady state concentration is\", Cout, \"[kg/m^3]\") "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4.3 Cascade configuration"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Problem Statement__\n",
    "\n",
    "500 l of fruit juice are concentrated in 5 h of steady state operation from an initial solid-residue content of 0.05 kg/l through four membrane separation units characterised by a total membrane area of 20 $\\mathrm{m^2}$ each.\n",
    "The flux of pure water through the membrane is captured by the empirical expression: $J=BC^{-1} [m/h]$ with $B=0.1$ where C is the solid residue concentration in kg/l.\n",
    "\n",
    "Is it more efficient to design a single stage configuration with four units in parallel or a cascade configuration with four units in series? \n",
    "\n",
    "<img src=\"cascade.png\" alt=\"Drawing\" style=\"width: 600px;\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Cascade: Solution trace__\n",
    "\n",
    "Each stage can be treated like a single unit in which the feed stream corresponds to the retentate stream from the previous unit.\n",
    "We can thus define: \n",
    "\n",
    "$$\n",
    "C_{OUT,i}=C_{IN,i+1}\n",
    "$$(PLYeq15)\n",
    "\n",
    "$$\n",
    "Q_{OUT,i}=Q_{IN,i+1}\n",
    "$$(PLYeq16)\n",
    "\n",
    "where the index $i$ identifies the stage. \n",
    "\n",
    "The material balances for each stage at steady state can thus be written as: \n",
    "\n",
    "$$\n",
    "\\frac{dV}{dt}=Q_{IN,i}-n_iJ(C_{IN,i+1})\\,A-Q_{IN,i+1}=0\n",
    "$$(PLYeq17)\n",
    "\n",
    "$$\n",
    "\\frac{dm}{dt}=Q_{IN,i}C_{IN}-Q_{IN,i+1}C_{IN,i+1}=0\n",
    "$$(PLYeq18)\n",
    "\n",
    "where $n_i$ is the number of modules used in stage $i$.\n",
    "\n",
    "For each stage we can thus compute the steady state concentration and concentrate volumetric flow by solving sequentially the following equations:  \n",
    "\n",
    "$$\n",
    "C_{IN,i+1}=C_{IN,i}+n_i\\frac{AB}{Q_{IN,i}}\n",
    "$$(PLYeq19)\n",
    "\n",
    "$$\n",
    "Q_{IN,i+1}=\\frac{C_{IN,i}{Q_{IN,i}}}{C_{IN,i+1}}\n",
    "$$(PLYeq20)\n",
    "\n",
    "with $i=1...N$ with $N$ is the total number of stages. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Numerical Solution__ \n",
    "\n",
    "The solution of a cascade composed of any number of stages, each formed by an arbitrary number of modules in parallel can be tackled sequentially through a simple cycle similar to the following, let's for example consider a system that reflects the sketch above i.e. with 4 stages implementing 4, 3, 2, and 1 modules each: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The steady state concentration is 720.7199999999999 [kg/m^3]\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt \n",
    "from scipy.optimize import fsolve\n",
    "\n",
    "# data: \n",
    "A=20;  \n",
    "B=0.1;\n",
    "C0=0.05*1E3; #kg /l * dm^3/m^3\n",
    "V0=500*1E-3; #l * m^3/dm^3\n",
    "process_time=5; # [h]\n",
    "\n",
    "\n",
    "# The number of elements of this array corresponds to the number of stages. \n",
    "# The value in each element is the number of modules per stage. \n",
    "n=np.array([4,3,2,1]);\n",
    "\n",
    "CIN=np.append(C0,np.zeros(np.size(n)-1))\n",
    "FIN=np.append(V0/process_time, np.zeros(np.size(n)-1));\n",
    "\n",
    "# Input to the intermediate stages\n",
    "for i in range(1,np.size(n)): \n",
    "    CIN[i]=CIN[i-1]+n[i-1]*A*B/FIN[i-1];\n",
    "    FIN[i]=CIN[i-1]*FIN[i-1]/CIN[i];\n",
    "\n",
    "# Output concentration\n",
    "Cout=CIN[np.size(n)-1]+n[np.size(n)-1]*A*B/FIN[np.size(n)-1];\n",
    "\n",
    "print(\"The steady state concentration is\", Cout, \"[kg/m^3]\") "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In order to answer the problem request one should solve the system for two different configurations. \n",
    "In the first, representing a single-stage configuration with four membrane units in parallel, the number of stages should be set to $N=1$ and the number of units in the first stage $n_1$ to 4. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The steady state concentration is 130.0 [kg/m^3]\n"
     ]
    }
   ],
   "source": [
    "# The number of elements of this array corresponds to the number of stages. \n",
    "# The value in each element is the number of modules per stage. \n",
    "n=np.array([4]);\n",
    "\n",
    "CIN=np.append(C0,np.zeros(np.size(n)-1))\n",
    "FIN=np.append(V0/process_time, np.zeros(np.size(n)-1));\n",
    "\n",
    "# Input to the intermediate stages\n",
    "for i in range(1,np.size(n)): \n",
    "    CIN[i]=CIN[i-1]+n[i-1]*A*B/FIN[i-1];\n",
    "    FIN[i]=CIN[i-1]*FIN[i-1]/CIN[i];\n",
    "\n",
    "# Output concentration\n",
    "Cout=CIN[np.size(n)-1]+n[np.size(n)-1]*A*B/FIN[np.size(n)-1];\n",
    "\n",
    "print(\"The steady state concentration is\", Cout, \"[kg/m^3]\") "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the second the number of stages should be set to $N=4$, each of the stages being assembled as a single unit ($n_i=1$ for $i=[1, 4]$)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The steady state concentration is 192.07999999999998 [kg/m^3]\n"
     ]
    }
   ],
   "source": [
    "# The number of elements of this array corresponds to the number of stages. \n",
    "# The value in each element is the number of modules per stage. \n",
    "n=np.array([1, 1, 1, 1]);\n",
    "\n",
    "CIN=np.append(C0,np.zeros(np.size(n)-1))\n",
    "FIN=np.append(V0/process_time, np.zeros(np.size(n)-1));\n",
    "\n",
    "# Input to the intermediate stages\n",
    "for i in range(1,np.size(n)): \n",
    "    CIN[i]=CIN[i-1]+n[i-1]*A*B/FIN[i-1];\n",
    "    FIN[i]=CIN[i-1]*FIN[i-1]/CIN[i];\n",
    "\n",
    "# Output concentration\n",
    "Cout=CIN[np.size(n)-1]+n[np.size(n)-1]*A*B/FIN[np.size(n)-1];\n",
    "\n",
    "print(\"The steady state concentration is\", Cout, \"[kg/m^3]\") "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The cascade configuration allows for a more efficient process since it allows to obtain a larger concentration with the same number of modules. "
   ]
  }
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