(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 32980, 936]*) (*NotebookOutlinePosition[ 33645, 959]*) (* CellTagsIndexPosition[ 33601, 955]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Liquid Liquid Equilibrium Using NDSolve\nCase of a ternary \ system: water - ethyl acetate - acetone at 15 \[Degree]C", FontSize->18], StyleBox["\nAuthor's Data: ", FontSize->16], StyleBox["Housam BINOUS\nDepartment of Chemical Engineering\nNational \ Institute of Applied Sciences and Technology\nTunis, TUNISIA\nEmail: \ binoushousam@yahoo.com ", FontSize->16, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}] }], "Title", TextAlignment->Center, Background->RGBColor[0.734386, 0.996109, 0.648447]], Cell[BoxData[ \(Off[General::"\"]\)], "Input"], Cell[CellGroupData[{ Cell[TextData[{ "water = component 1\nethylacetate = component 2\nacetone = component 3\n\n\ ", StyleBox["Activities using the NRTL model", FontWeight->"Bold"] }], "Subsubtitle", Background->RGBColor[0.996109, 0.996109, 0.621103]], Cell[CellGroupData[{ Cell[BoxData[ \(GAMMA1 = Exp[\((t21/\((R\ T)\)\ G21\ x2 + t31/\((R\ T)\)\ G31\ x3)\)/\((x1 + G21\ x2 + G31\ x3)\) + x1/\((x1 + G21\ x2 + G31\ x3)\)\ \((\(-\((x2\ t21/\((R\ T)\)\ G21 + x3\ t31/\((R\ T)\)\ G31)\)\)/\((x1 + G21\ x2 + G31\ x3)\))\) + x2\ G12/\((G12\ x1 + x2 + G32\ x3)\)\ \((t12/\((R\ T)\) - \((x1\ t12/\((R\ T)\)\ G12 \ + x3\ t32/\((R\ T)\)\ G32)\)/\((G12\ x1 + x2 + G32\ x3)\))\) + \[IndentingNewLine]x3\ G13/\((G13\ x1 + G23\ x2\ + x3)\)\ \((t13/\((R\ T)\) - \((x1\ t13/\((R\ T)\)\ G13 + x2\ t23/\((R\ T)\)\ G23)\)/\((G13\ x1 + G23\ x2\ + x3)\))\)]\)], "Input"], Cell[BoxData[ \(\[ExponentialE]\^\(\(x1\ \((\(-\(\(G21\ t21\ x2\)\/\(R\ T\)\)\) - \(G31\ \ t31\ x3\)\/\(R\ T\))\)\)\/\((x1 + G21\ x2 + G31\ x3)\)\^2 + \(\(G21\ t21\ \ x2\)\/\(R\ T\) + \(G31\ t31\ x3\)\/\(R\ T\)\)\/\(x1 + G21\ x2 + G31\ x3\) + \ \(G13\ x3\ \((t13\/\(R\ T\) - \(\(G13\ t13\ x1\)\/\(R\ T\) + \(G23\ t23\ x2\)\ \/\(R\ T\)\)\/\(G13\ x1 + G23\ x2 + x3\))\)\)\/\(G13\ x1 + G23\ x2 + x3\) + \ \(G12\ x2\ \((t12\/\(R\ T\) - \(\(G12\ t12\ x1\)\/\(R\ T\) + \(G32\ t32\ x3\)\ \/\(R\ T\)\)\/\(G12\ x1 + x2 + G32\ x3\))\)\)\/\(G12\ x1 + x2 + G32\ \ x3\)\)\)], "Output"], Cell[CellGroupData[{ Cell[BoxData[ \(GAMMA1p = Exp[\((t21/\((R\ T)\)\ G21\ xp2 + t31/\((R\ T)\)\ G31\ xp3)\)/\((xp1 + G21\ xp2 + G31\ xp3)\) + xp1/\((xp1 + G21\ xp2 + G31\ xp3)\)\ \((\(-\((xp2\ t21/\((R\ T)\)\ G21 + xp3\ t31/\((R\ T)\)\ G31)\)\)/\((xp1 + G21\ xp2 + G31\ xp3)\))\) + xp2\ G12/\((G12\ xp1 + xp2 + G32\ xp3)\)\ \((t12/\((R\ T)\) - \((xp1\ t12/\((R\ T)\)\ \ G12 + xp3\ t32/\((R\ T)\)\ G32)\)/\((G12\ xp1 + xp2 + G32\ xp3)\))\) + \[IndentingNewLine]xp3\ G13/\((G13\ \ xp1 + G23\ xp2\ + xp3)\)\ \((t13/\((R\ T)\) - \((xp1\ t13/\((R\ T)\)\ G13 + xp2\ t23/\((R\ T)\)\ G23)\)/\((G13\ xp1 + G23\ xp2\ + xp3)\))\)]\)], "Input"], Cell[BoxData[ \(\[ExponentialE]\^\(\(xp1\ \((\(-\(\(G21\ t21\ xp2\)\/\(R\ T\)\)\) - \ \(G31\ t31\ xp3\)\/\(R\ T\))\)\)\/\((xp1 + G21\ xp2 + G31\ xp3)\)\^2 + \ \(\(G21\ t21\ xp2\)\/\(R\ T\) + \(G31\ t31\ xp3\)\/\(R\ T\)\)\/\(xp1 + G21\ \ xp2 + G31\ xp3\) + \(G13\ xp3\ \((t13\/\(R\ T\) - \(\(G13\ t13\ xp1\)\/\(R\ T\ \) + \(G23\ t23\ xp2\)\/\(R\ T\)\)\/\(G13\ xp1 + G23\ xp2 + xp3\))\)\)\/\(G13\ \ xp1 + G23\ xp2 + xp3\) + \(G12\ xp2\ \((t12\/\(R\ T\) - \(\(G12\ t12\ xp1\)\ \/\(R\ T\) + \(G32\ t32\ xp3\)\/\(R\ T\)\)\/\(G12\ xp1 + xp2 + G32\ \ xp3\))\)\)\/\(G12\ xp1 + xp2 + G32\ xp3\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(GAMMA2 = Exp[\((t12/\((R\ T)\)\ G12\ x1 + t32/\((R\ T)\)\ G32\ x3)\)/\((G12\ x1 + x2 + G32\ x3)\) + x1\ G21/\((x1 + G21\ x2 + G31\ x3)\)\ \((t21/\((R\ T)\) - \((x2\ t21/\((R\ T)\)\ G21 \ + x3\ t31/\((R\ T)\)\ G31)\)/\((x1 + G21\ x2 + G31\ x3)\))\) + x2/\((G12\ x1 + x2 + G32\ x3)\)\ \((\(-\((x1\ t12/\((R\ T)\)\ G12 + x3\ t32/\((R\ T)\)\ G32)\)\)/\((G12\ x1 + x2 + G32\ x3)\))\) + x3\ G23/\((G13\ x1 + G23\ x2\ + x3)\)\ \((t23/\((R\ T)\) - \((x1\ t13/\((R\ T)\)\ G13 + x2\ t23/\((R\ T)\)\ G23)\)/\((G13\ x1 + G23\ x2\ + x3)\))\)]\)], "Input"], Cell[BoxData[ \(\[ExponentialE]\^\(\(x2\ \((\(-\(\(G12\ t12\ x1\)\/\(R\ T\)\)\) - \(G32\ \ t32\ x3\)\/\(R\ T\))\)\)\/\((G12\ x1 + x2 + G32\ x3)\)\^2 + \(\(G12\ t12\ \ x1\)\/\(R\ T\) + \(G32\ t32\ x3\)\/\(R\ T\)\)\/\(G12\ x1 + x2 + G32\ x3\) + \ \(G23\ x3\ \((t23\/\(R\ T\) - \(\(G13\ t13\ x1\)\/\(R\ T\) + \(G23\ t23\ x2\)\ \/\(R\ T\)\)\/\(G13\ x1 + G23\ x2 + x3\))\)\)\/\(G13\ x1 + G23\ x2 + x3\) + \ \(G21\ x1\ \((t21\/\(R\ T\) - \(\(G21\ t21\ x2\)\/\(R\ T\) + \(G31\ t31\ x3\)\ \/\(R\ T\)\)\/\(x1 + G21\ x2 + G31\ x3\))\)\)\/\(x1 + G21\ x2 + G31\ \ x3\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(GAMMA2p = Exp[\((t12/\((R\ T)\)\ G12\ xp1 + t32/\((R\ T)\)\ G32\ xp3)\)/\((G12\ xp1 + xp2 + G32\ xp3)\) + xp1\ G21/\((xp1 + G21\ xp2 + G31\ xp3)\)\ \((t21/\((R\ T)\) - \((xp2\ t21/\((R\ T)\)\ \ G21 + xp3\ t31/\((R\ T)\)\ G31)\)/\((xp1 + G21\ xp2 + G31\ xp3)\))\) + xp2/\((G12\ xp1 + xp2 + G32\ xp3)\)\ \((\(-\((xp1\ t12/\((R\ T)\)\ G12 + xp3\ t32/\((R\ T)\)\ G32)\)\)/\((G12\ xp1 + xp2 + G32\ xp3)\))\) + xp3\ G23/\((G13\ xp1 + G23\ xp2\ + xp3)\)\ \((t23/\((R\ T)\) - \((xp1\ t13/\((R\ T)\)\ G13 + xp2\ t23/\((R\ T)\)\ G23)\)/\((G13\ xp1 + G23\ xp2\ + xp3)\))\)]\)], "Input"], Cell[BoxData[ \(\[ExponentialE]\^\(\(xp2\ \((\(-\(\(G12\ t12\ xp1\)\/\(R\ T\)\)\) - \ \(G32\ t32\ xp3\)\/\(R\ T\))\)\)\/\((G12\ xp1 + xp2 + G32\ xp3)\)\^2 + \ \(\(G12\ t12\ xp1\)\/\(R\ T\) + \(G32\ t32\ xp3\)\/\(R\ T\)\)\/\(G12\ xp1 + \ xp2 + G32\ xp3\) + \(G23\ xp3\ \((t23\/\(R\ T\) - \(\(G13\ t13\ xp1\)\/\(R\ T\ \) + \(G23\ t23\ xp2\)\/\(R\ T\)\)\/\(G13\ xp1 + G23\ xp2 + xp3\))\)\)\/\(G13\ \ xp1 + G23\ xp2 + xp3\) + \(G21\ xp1\ \((t21\/\(R\ T\) - \(\(G21\ t21\ xp2\)\ \/\(R\ T\) + \(G31\ t31\ xp3\)\/\(R\ T\)\)\/\(xp1 + G21\ xp2 + G31\ \ xp3\))\)\)\/\(xp1 + G21\ xp2 + G31\ xp3\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(GAMMA3 = Exp[\((t13/\((R\ T)\)\ G13\ x1 + t23/\((R\ T)\)\ G23\ x2)\)/\((G13\ x1 + G23\ x2 + x3)\) + x1\ G31/\((x1 + G21\ x2 + G31\ x3)\)\ \((t31/\((R\ T)\) - \((x2\ t21/\((R\ T)\)\ G21 \ + x3\ t31/\((R\ T)\)\ G31)\)/\((x1 + G21\ x2 + G31\ x3)\))\) + x2\ G32/\((G12\ x1 + x2 + G32\ x3)\)\ \((t32/\((R\ T)\) - \((x1\ t12/\((R\ T)\)\ G12 \ + x3\ t32/\((R\ T)\)\ G32)\)/\((G12\ x1 + x2 + G32\ x3)\))\) + x3/\((G13\ x1 + G23\ x2\ + x3)\) \((\(-\((x1\ t13/\((R\ T)\)\ G13 + x2\ t23/\((R\ T)\)\ G23)\)\)/\((G13\ x1 + G23\ x2\ + x3)\))\)]\)], "Input"], Cell[BoxData[ \(\[ExponentialE]\^\(\(\((\(-\(\(G13\ t13\ x1\)\/\(R\ T\)\)\) - \(G23\ \ t23\ x2\)\/\(R\ T\))\)\ x3\)\/\((G13\ x1 + G23\ x2 + x3)\)\^2 + \(\(G13\ t13\ \ x1\)\/\(R\ T\) + \(G23\ t23\ x2\)\/\(R\ T\)\)\/\(G13\ x1 + G23\ x2 + x3\) + \ \(G31\ x1\ \((t31\/\(R\ T\) - \(\(G21\ t21\ x2\)\/\(R\ T\) + \(G31\ t31\ x3\)\ \/\(R\ T\)\)\/\(x1 + G21\ x2 + G31\ x3\))\)\)\/\(x1 + G21\ x2 + G31\ x3\) + \ \(G32\ x2\ \((t32\/\(R\ T\) - \(\(G12\ t12\ x1\)\/\(R\ T\) + \(G32\ t32\ x3\)\ \/\(R\ T\)\)\/\(G12\ x1 + x2 + G32\ x3\))\)\)\/\(G12\ x1 + x2 + G32\ \ x3\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(GAMMA3p = Exp[\((t13/\((R\ T)\)\ G13\ xp1 + t23/\((R\ T)\)\ G23\ xp2)\)/\((G13\ xp1 + G23\ xp2 + xp3)\) + xp1\ G31/\((xp1 + G21\ xp2 + G31\ xp3)\)\ \((t31/\((R\ T)\) - \((xp2\ t21/\((R\ T)\)\ \ G21 + xp3\ t31/\((R\ T)\)\ G31)\)/\((xp1 + G21\ xp2 + G31\ xp3)\))\) + xp2\ G32/\((G12\ xp1 + xp2 + G32\ xp3)\)\ \((t32/\((R\ T)\) - \((xp1\ t12/\((R\ T)\)\ \ G12 + xp3\ t32/\((R\ T)\)\ G32)\)/\((G12\ xp1 + xp2 + G32\ xp3)\))\) + xp3/\((G13\ xp1 + G23\ xp2\ + xp3)\) \((\(-\((xp1\ t13/\((R\ T)\)\ G13 + xp2\ t23/\((R\ T)\)\ G23)\)\)/\((G13\ xp1 + G23\ xp2\ + xp3)\))\)]\)], "Input"], Cell[BoxData[ \(\[ExponentialE]\^\(\(\((\(-\(\(G13\ t13\ xp1\)\/\(R\ T\)\)\) - \(G23\ \ t23\ xp2\)\/\(R\ T\))\)\ xp3\)\/\((G13\ xp1 + G23\ xp2 + xp3)\)\^2 + \(\(G13\ \ t13\ xp1\)\/\(R\ T\) + \(G23\ t23\ xp2\)\/\(R\ T\)\)\/\(G13\ xp1 + G23\ xp2 + \ xp3\) + \(G31\ xp1\ \((t31\/\(R\ T\) - \(\(G21\ t21\ xp2\)\/\(R\ T\) + \(G31\ \ t31\ xp3\)\/\(R\ T\)\)\/\(xp1 + G21\ xp2 + G31\ xp3\))\)\)\/\(xp1 + G21\ xp2 \ + G31\ xp3\) + \(G32\ xp2\ \((t32\/\(R\ T\) - \(\(G12\ t12\ xp1\)\/\(R\ T\) + \ \(G32\ t32\ xp3\)\/\(R\ T\)\)\/\(G12\ xp1 + xp2 + G32\ xp3\))\)\)\/\(G12\ xp1 \ + xp2 + G32\ xp3\)\)\)], "Output"] }, Open ]], Cell[BoxData[ \(G12 = Exp[\(-a12\)\ t12/\((R\ T)\)]; G21 = Exp[\(-a12\)\ t21/\((R\ T)\)]; G13 = Exp[\(-a13\)\ t13/\((R\ T)\)]; G31 = Exp[\(-a13\)\ t31/\((R\ T)\)]; G32 = Exp[\(-a32\)\ t32/\((R\ T)\)]; G23 = Exp[\(-a32\)\ t23/\((R\ T)\)];\)], "Input"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Gas constant, temperature and binary interaction parameters for the NRTL \ model\ \>", "Subsubtitle", Background->RGBColor[0.996109, 0.996109, 0.621103]], Cell[BoxData[{ \(\(R = 1.987;\)\), "\[IndentingNewLine]", \(a13 = 0.585600018501282; a12 = 0.410400003194809;\), "\[IndentingNewLine]", \(\(a32 = 0.291200011968613;\)\), "\[IndentingNewLine]", \(t31 = 750.318115234375; t13 = 1299.39501953125;\), "\[IndentingNewLine]", \(t23 = 467.125701904297; t32 = \(-304.988403320313\);\), "\[IndentingNewLine]", \(\(t12 = 2316.36303710938;\)\), "\[IndentingNewLine]", \(\(t21 = 935.687927246094;\)\)}], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(T = 273.15 + 15\)], "Input"], Cell[BoxData[ \(288.15`\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Computing a Tie line with FindRoot", "Subsubtitle", Background->RGBColor[0.996109, 0.996109, 0.621103]], Cell[BoxData[ \(X1 = 0.7; X2 = 0.2;\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(sol[1] = FindRoot[{\((GAMMA1\ x1 \[Equal] \ GAMMA1p\ xp1)\), \((GAMMA2\ x2 \[Equal] \ GAMMA2p\ xp2)\), \[IndentingNewLine]\((GAMMA3\ x3 \[Equal] \ GAMMA3p\ xp3)\), \[IndentingNewLine]\((x1\ L + xp1\ Lp \[Equal] \ X1)\), \((x2\ L + xp2\ Lp \[Equal] \ X2)\), \((1 \[Equal] \ xp1 + xp2 + xp3)\), \((1 \[Equal] \ x1 + x2 + x3)\), \((L + Lp \[Equal] 1)\)}, {x1, 0.95}, {x2, 0.01}, {xp1, 0.5}, {xp2, 0.4}, {L, 0.05}, {Lp, 0.04}, {x3, 0.01}, {xp3, 0.1}, \n\tMaxIterations \[Rule] 10000]\)], "Input"], Cell[BoxData[ \({x1 \[Rule] 0.9668685582716752`, x2 \[Rule] 0.012388890427507862`, xp1 \[Rule] 0.5740880749385077`, xp2 \[Rule] 0.2885172690337957`, L \[Rule] 0.32056563501575575`, Lp \[Rule] 0.6794343649842443`, x3 \[Rule] 0.02074255130081697`, xp3 \[Rule] 0.13739465602769668`}\)], "Output"] }, Open ]] }, Closed]] }, Open ]], Cell["Computing a Tie line with NDSolve", "Subsubtitle", Background->RGBColor[0, 1, 1]], Cell[CellGroupData[{ Cell["Coordinates of the first point on the tie line.", "Subsubtitle", Background->RGBColor[0.996109, 0.996109, 0.621103]], Cell[CellGroupData[{ Cell[BoxData[ \(x1 = \(sol[1]\)[\([1, 2]\)]\)], "Input"], Cell[BoxData[ \(0.9668685582716752`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(x2 = \(sol[1]\)[\([2, 2]\)]\)], "Input"], Cell[BoxData[ \(0.012388890427507862`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(x3 = 1 - x1 - x2\)], "Input"], Cell[BoxData[ \(0.020742551300816967`\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Using NDSolve to look for the steady state, which corresponds to the solution \ of the nonlinear algebraic equations.\ \>", "Subsubtitle", Background->RGBColor[0.996109, 0.996109, 0.621103]], Cell[CellGroupData[{ Cell[BoxData[ \(sol1 = NDSolve[{D[ xp1[t], {t, 1}] \[Equal] \ \((\(\(GAMMA1\ x1 - \ GAMMA1p\ xp1 /. xp1 \[Rule] xp1[t]\) /. xp2 \[Rule] \ xp2[t]\) /. xp3 \[Rule] \ xp3[t])\), \[IndentingNewLine]D[ xp2[t], {t, 1}] \[Equal] \ \((\(\(GAMMA2\ x2 - \ GAMMA2p\ xp2 /. xp1 \[Rule] xp1[t]\) /. xp2 \[Rule] \ xp2[t]\) /. xp3 \[Rule] \ xp3[t])\), \[IndentingNewLine]D[ xp3[t], {t, 1}] \[Equal] \ \((\(\(GAMMA3\ x3 - \ GAMMA3p\ xp3 /. xp1 \[Rule] xp1[t]\) /. xp2 \[Rule] \ xp2[t]\) /. xp3 \[Rule] \ xp3[t])\), \[IndentingNewLine]xp1[0] \[Equal] \ 0.4, xp2[0] \[Equal] \ 0.2, xp3[0] \[Equal] \ 0.1}, {xp1, xp2, xp3}, {t, 0, 1000}]\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{"xp1", "\[Rule]", TagBox[\(InterpolatingFunction[{{0.`, 1000.`}}, "<>"]\), False, Editable->False]}], ",", RowBox[{"xp2", "\[Rule]", TagBox[\(InterpolatingFunction[{{0.`, 1000.`}}, "<>"]\), False, Editable->False]}], ",", RowBox[{"xp3", "\[Rule]", TagBox[\(InterpolatingFunction[{{0.`, 1000.`}}, "<>"]\), False, Editable->False]}]}], "}"}], "}"}]], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Coordinates of the second point on the tie line are the steady state solution \ (obtained at t=1000). 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