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    "# 1.11 – Practice Problem 1\n",
    "\n",
    "## Question\n",
    "\n",
    "In wastewater treatment, an important step in the process is sedimentation. Sedimentation is the process of allowing particulate matter in a fluid to settle to the bottom of a tank. In the sedimentation step, a coagulant can be used to coagulate bacteria and fine particles to allow the sedimentation to occur faster. The most common molecular species used for coagulation is alum, $Al_2 (SO_4)_3 \\cdot 14H_2 O$.\n",
    "\n",
    "You are a chemical engineer working in a waste water treatment plant. You are asked to make a material balance on the sedimentation process. Waste water flows into a sedimentation tank at 200 kg/s. The waste water contains 0.01 wt% of bacteria, 2 wt% particulate matter, and the balance pure water. Pure alum is added to the sedimentation tank in a separate stream. Alum allows 80% of the bacteria and 100 % of the particulate matter to settle. Just enough alum is added for this ratio of settlement. The sludge at the bottom of the tank exits the tank with 20 wt% of water and 40 wt% of particulate matter. The semi-treated water exits the tank in a separate stream.\n",
    "\n",
    "a) Draw and label a block flow diagram (BFD).\n",
    "\n",
    "b) Perform a degree of freedom analysis.\n",
    "\n",
    "c) Fully balance the remainder of the block flow diagram.\n",
    "\n",
    "## Answer\n",
    "\n",
    "### a) Draw and label a block flow diagram (BFD)\n",
    "\n",
    "First, let's draw the basic layout of the BFD:\n",
    "\n",
    "![](../figures/Module-1/Sedimentation-2.png)\n",
    "\n",
    "Attribution: Said Zaid-Alkailani & UBC [CC BY 4.0 de (https://creativecommons.org/licenses/by/4.0/)]\n",
    "\n",
    "Using the information given, let's fill out as much of the BFD as we can.\n",
    "\n",
    "$$ x_{(1, \\space H_2 O)} = 1 - x_{(1, \\space Particulate)} - x_{(1, \\space Bacteria)} = 1 - 0.02 - 0.0001 = 0.9799 $$\n",
    "\n",
    "$$ x_{(2, \\space Al_2 (SO_4)_3 \\cdot 14 H_2 O)} = 1 $$\n",
    "\n",
    "$$ x_{(4, \\space H_2 O)} = 0.20 $$\n",
    "\n",
    "$$ x_{(4, \\space Particulate)} = 0.40 $$\n",
    "\n",
    "![](../figures/Module-1/Sedimentation-3.png)\n",
    "\n",
    "Attribution: Said Zaid-Alkailani & UBC [CC BY 4.0 de (https://creativecommons.org/licenses/by/4.0/)]\n",
    "\n",
    "### b) Perform a degree of freedom analysis\n",
    "\n",
    "We have 7 unknowns, so we need to find 7 equations. First lets do all of the mass balances:\n",
    "\n",
    "$$ \\dot{m}_1 x_{(1, \\space H_2 O)} = \\dot{m}_3 x_{(3, \\space H_2 O)} + \\dot{m}_4 x_{(4, \\space H_2 O)} $$\n",
    "\n",
    "$$ \\dot{m}_1 x_{(1, Particulate)} = \\dot{m}_4 x_{(4, Particulate)}$$\n",
    "\n",
    "$$ \\dot{m}_1 x_{(1, Bacteria)} = \\dot{m}_3 x_{(3, Bacteria)} + \\dot{m}_4 x_{(4, Bacteria)}$$\n",
    "\n",
    "$$ \\dot{m}_2 x_{(2, \\space Al_2 (SO_4)_3 \\cdot 14 H_2 O)} = \\dot{m}_4 x_{(4, \\space Al_2 (SO_4)_3 \\cdot 14 H_2 O)} $$\n",
    "\n",
    "Now lets do the mass fraction balance:\n",
    "\n",
    "$$ x_{(3, \\space H_2 O)} = 1 - x_{(3, Bacteria)} $$\n",
    "\n",
    "$$ x_{(4, \\space Particulate)} = 1 - x_{(4, H_2 O)} - x_{(4, Bacteria)} - x_{(4, \\space Al_2 (SO_4)_3 \\cdot 14 H_2 O)} $$\n",
    "\n",
    "Next lets write our the extra equation:\n",
    "\n",
    "$$ 0.80 \\cdot \\dot{m}_1 x_{(1, Bacteria)} = \\dot{m}_4 x_{(4, Bacteria)} $$\n",
    "\n",
    "Finally we can solve for the degrees of freedom:\n",
    "\n",
    "$$ \\text{DOF} = 7 - 7 = 0 $$\n",
    "\n",
    "### c) Fully balance the remainder of the block flow diagram\n",
    "\n",
    "To fully we balance the diagram we must first solve for all of the unknowns.\n",
    "\n",
    "#### 1. Solve for $\\dot{m}_4$ using particulate balance:\n",
    "\n",
    "$$ \\dot{m}_1 x_{(1, Particulate)} = \\dot{m}_4 x_{(4, Particulate)}$$\n",
    "\n",
    "$$ \\dot{m}_4 = \\frac{\\dot{m}_1 x_{(1, Particulate)}}{x_{(4, Particulate)}} = \\frac{(200)(0.02)}{0.4} \\space \\frac{kg}{s} = 10 \\space \\frac{kg}{s} $$\n",
    "\n",
    "#### 2. Solve for $x_{(4, Bacteria)}$ using the extra equation:\n",
    "\n",
    "$$ 0.80 \\cdot \\dot{m}_1 x_{(1, Bacteria)} = \\dot{m}_4 x_{(4, Bacteria)} $$\n",
    "\n",
    "$$ x_{(4, Bacteria)} = \\frac{0.80 \\cdot \\dot{m}_1 x_{(1, Bacteria)}}{\\dot{m}_4}  = \\frac{\\big( 0.80 \\big) \\big( 200 \\space \\frac{kg}{s} \\big) \\big(0.0001 \\big)}{10 \\space \\frac{kg}{s}} = 0.0016$$\n",
    "\n",
    "#### 3. Solve for $x_{(4, \\space Al_2 (SO_4)_3 \\cdot 14 H_2 O)}$ using the fractional mass balance:\n",
    "\n",
    "$$ x_{(4, \\space Al_2 (SO_4)_3 \\cdot 14 H_2 O)} = 1 - x_{(4, H_2 O)} - x_{(4, Bacteria)} - x_{(4, \\space Particulate)} = 1 - 0.20 - 0.0016 - 0.40 = 0.3984 $$\n",
    "\n",
    "#### 4. Solve for $\\dot{m}_2$ using the alum balance:\n",
    "\n",
    "$$ \\dot{m}_2 x_{(2, \\space Al_2 (SO_4)_3 \\cdot 14 H_2 O)} = \\dot{m}_4 x_{(4, \\space Al_2 (SO_4)_3 \\cdot 14 H_2 O)} = \\bigg(10 \\space \\frac{kg}{s} \\bigg) \\bigg(0.3984 \\bigg) = 3.984 \\space \\frac{kg}{s}$$\n",
    "\n",
    "\n",
    "#### 5. Solve for $\\dot{m}_3$ using a combination of the water balance and bacteria balance:\n",
    "\n",
    "$$ \\dot{m}_1 x_{(1, Bacteria)} = \\dot{m}_3 x_{(3, Bacteria)} + \\dot{m}_4 x_{(4, Bacteria)}$$\n",
    "\n",
    "$$ \\dot{m}_3 = \\frac{ \\dot{m}_1 x_{(1, \\space Bacteria)} - \\dot{m}_4 x_{(4, \\space Bacteria)}}{x_{(3, \\space Bacteria)}}$$\n",
    "\n",
    "$$ \\dot{m}_1 x_{(1, \\space H_2 O)} = \\dot{m}_3 x_{(3, \\space H_2 O)} + \\dot{m}_4 x_{(4, \\space H_2 O)} $$\n",
    "\n",
    "$$ \\dot{m}_3 = \\frac{ \\dot{m}_1 x_{(1, H_2 O)} - \\dot{m}_4 x_{(4, H_2 O)}}{x_{(3, H_2 O)}}$$\n",
    "\n",
    "We can equate these two equations to get:\n",
    "\n",
    "$$ \\frac{ \\dot{m}_1 x_{(1, \\space Bacteria)} - \\dot{m}_4 x_{(4, \\space Bacteria)}}{x_{(3, \\space Bacteria)}} = \\frac{ \\dot{m}_1 x_{(1, H_2 O)} - \\dot{m}_4 x_{(4, H_2 O)}}{x_{(3, H_2 O)}}$$\n",
    "\n",
    "and using the fractional mass balance\n",
    "\n",
    "$$x_{(3, \\space H_2 O)} = 1 - x_{(3, \\space Bacteria)}$$\n",
    "\n",
    "we can solve for $x_{(3, \\space Bacteria)}$\n",
    "\n",
    "$$ \\frac{ \\dot{m}_1 x_{(1, \\space Bacteria)} - \\dot{m}_4 x_{(4, \\space Bacteria)}}{x_{(3, \\space Bacteria)}} = \\frac{ \\dot{m}_1 x_{(1, H_2 O)} - \\dot{m}_4 x_{(4, H_2 O)}}{1 - x_{(3, \\space Bacteria)}}$$\n",
    "\n",
    "$$ \\frac{0.004}{x_{(3, \\space Bacteria)}} = \\frac{193.98}{1 - x_{(3, \\space Bacteria)}}$$\n",
    "\n",
    "$$ x_{(3, \\space Bacteria)} = 0.00002 $$\n",
    "\n",
    "$$ \\therefore \\space x_{(3, \\space H_2 O)} = 1 - 0.00002 = 0.99998 $$\n",
    "\n",
    "$$ \\therefore \\space \\dot{m}_3 = \\frac{ \\dot{m}_1 x_{(1, H_2 O)} - \\dot{m}_4 x_{(4, H_2 O)}}{x_{(3, H_2 O)}} = \\frac{ 193.98 }{0.99998} \\space \\frac{kg}{s} = 193.98 \\space \\frac{kg}{s}$$\n",
    "\n",
    "Now that we have completely solved the material balance, we can complete our BFD:\n",
    "\n",
    "![](../figures/Module-1/Sedimentation-4.png)\n",
    "\n",
    "Attribution: Said Zaid-Alkailani & UBC [CC BY 4.0 de (https://creativecommons.org/licenses/by/4.0/)]"
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