(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. 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). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. 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[ 455591, 18429]*) (*NotebookOutlinePosition[ 456507, 18458]*) (* CellTagsIndexPosition[ 456463, 18454]*) (*WindowFrame->Normal*) Notebook[{ Cell[BoxData[ StyleBox[\(Calculating\ 1 D\ Concentration\ Profiles\ of\ Melt\ Zone\ Refined\ Materials\), "Title"]], "Input"], Cell[CellGroupData[{ Cell["\<\ Scott Beckman January 2003 University of California at Berkeley Department of Materials Science and Engineering 577 Evans Hall #1760 Berkeley, CA 94720 sbeckman@uclink.berkeley.edu\ \>", "Subsubtitle"], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell["\<\ The basic idea of melt zone refining is that you have a large bar \ of material with impurities. You then past a liquid region down the length \ of the bar. Because the impurities have a higher solubility in liquid than \ in the solid, as the liquid region moves the length of the bar the impurities \ become trapped in the liquid and are moved toward the end of the bar. By \ repeatedly passing this narrow melted region the impurities are eventually \ pulled to one end of the bar. After enough passes a final distribution is \ reached. The shape of this region is exponential and the values depend on \ the initial concentration in the bar, the width of the melt region (relative \ to the total bar length), and the segregation coefficient. Below I step \ through how to calculate the defect distribution during this process. \ \>", \ "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Parameters", "Section"], Cell[TextData[{ "Initial, uniform, concentration of defects in bar, in units ", Cell[BoxData[ \(TraditionalForm\`cm\^\(-3\)\)]], ". " }], "Text"], Cell[BoxData[ \(\(co = 1*^20;\)\)], "Input"], Cell[TextData[{ "Segregation coefficient, ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(k\_o\), "TraditionalForm"], "=", \(C\_solid\/C\_liquid\)}], TraditionalForm]]] }], "Text"], Cell[BoxData[ \(\(k0 = 0.35;\)\)], "Input"], Cell[TextData[{ "Geometry of bar: length=100cm, area=\[Pi] ", Cell[BoxData[ \(TraditionalForm\`\((5 cm)\)\^2\)]] }], "Text"], Cell[BoxData[{ \(\(length = 100;\)\), "\[IndentingNewLine]", \(\(area = N[\(5\^2\) \[Pi]];\)\)}], "Input"], Cell["Heater length, 3cm", "Text"], Cell[BoxData[ \(\(meltlength = 3;\)\)], "Input"], Cell["\<\ Number of subdivisions of total bar length, this needs to be large \ enough that the number of steps that are liquid (meltsteps) are small \ compared to the total number of steps (nsteps)\ \>", "Text"], Cell[BoxData[ \(\(nsteps = 500;\)\)], "Input"], Cell["\<\ Number of subdivisions in melt region. 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ool>0?ooo`00403oool00`000000oooo0?ooo`3o0?ooo`h0oooo000@0?ooo`030000003oool0oooo 0?l0oooo3P3oool00100oooo00<000000?ooo`3oool0o`3oool>0?ooo`00403oool00`000000oooo 0?ooo`3o0?ooo`h0oooo000@0?ooo`030000003oool0oooo0?l0oooo3P3oool00001\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-32.4268, -7.11984, 1.92514, \ 0.573794}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell["\<\ The heater will step along the bar, one step at a time starting at \ step=1. \ \>", "Text"], Cell[BoxData[ \(\(step = 1;\)\)], "Input"], Cell["\<\ The number of impurities in the melted region are given by the sum \ of the impurities in each melted region. The impurity in a region is the \ concentration of impurities in the region times the volume of the region. \ \ \>", "Text"], Cell[BoxData[ \(\(liquidimpurity = Sum[concentration[\([q]\)], {q, step, step + meltsteps - 1}]* stepvol;\)\)], "Input"], Cell["\<\ The concentration in the liquid region is the number of impurities \ in the liquid region divided by the volume of the liquid region\ \>", "Text"], Cell[BoxData[ \(\(liquidcon = liquidimpurity/\((meltsteps*stepvol)\);\)\)], "Input"], Cell[BoxData[ \(When\ the\ heater\ advances\ a\ step\ the\ concentration\ of\ \ impurities\ that\ solidify\ into\ the\ solid\ are\ given\ by\ k\_o\)], "Text",\ FontFamily->"Times"], Cell[BoxData[ \(\(freezecon = liquidcon*k0;\)\)], "Input"], Cell["Write this to the concentration profile", "Text"], Cell[BoxData[ \(\(concentration = ReplacePart[concentration, freezecon, step];\)\)], "Input"], Cell["\<\ When the liquid freezes impurities will be ejected into the liquid. \ This will increase the concentration of impurities in the liquid. \ \>", \ "Text"], Cell["\<\ The number of impurities absorbed into the solid is given by the \ concentration in the solid times the volume of the newly frozen region\ \>", \ "Text"], Cell[BoxData[ \(\(freezimpurity = freezecon*stepvol;\)\)], "Input"], Cell["\<\ The number of impurities in the diminished liquid region is the \ number of initial number of impurities minus those absorbed by the \ solid\ \>", "Text"], Cell[BoxData[ \(\(liquidimpurity = liquidimpurity - freezimpurity;\)\)], "Input"], Cell["\<\ The concentration in the liquid is the number of impurities divided \ the smaller liquid region.\ \>", "Text"], Cell[BoxData[ \(\(liquidcon = liquidimpurity/\((\((meltsteps - 1)\)*stepvol)\);\)\)], "Input"], Cell["Write this to the concentration profile", "Text"], Cell[BoxData[ \(Do[\(concentration = ReplacePart[concentration, liquidcon, j];\)\[IndentingNewLine], {j, step + 1, step + meltsteps - 1}]\)], "Input"], Cell[CellGroupData[{ 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In normal freezing \ the solidification front moves forward one step at a time, ejecting solid \ into the liquid. No new solid is melted to dilute the liquid so as the final \ liquid solidifies the concentration goes to infinity. Since this bar is \ discrete, the final step is finite, but none the less, the curvature shows a \ trend toward infinity. \ \>", "Text"], Cell["The impurity in the liquid is given by ", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(step = nsteps - meltsteps + 1\)], "Input"], Cell[BoxData[ \(486\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(liquidimpurity = Sum[concentration[\([q]\)], {q, step, nsteps}]*stepvol\)], "Input"], Cell[BoxData[ \(6.731937694389268`*^22\)], "Output"] }, Open ]], Cell["The concentration in the liquid is then", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(liquidcon = liquidimpurity/\((\((nsteps - step)\)*stepvol)\)\)], "Input"], Cell[BoxData[ \(3.061203316139169`*^20\)], "Output"] }, Open ]], Cell["The concentration of the frozen solid will be", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(freezecon = liquidcon*k0\)], "Input"], Cell[BoxData[ \(1.071421160648709`*^20\)], "Output"] }, Open ]], Cell["and will have a total number of impurities", 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