| 1 | <!doctype html public "-//w3c//dtd html 4.0 transitional//en"> |
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| 2 | <html> |
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| 3 | <head> |
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| 4 | <meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1"> |
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| 5 | <meta name="Author" content="Adam C. Powell, IV"> |
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| 6 | <meta name="GENERATOR" content="Mozilla/4.75 [en] (X11; U; Linux 2.2.17 i686) [Netscape]"> |
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| 7 | <title>Sphlow Flotation Calculator 0.2 by Adam Powell</title> |
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| 8 | </head> |
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| 9 | <body text="#000066" bgcolor="#CCFFFF" link="#003300" vlink="#330033" alink="#990000"> |
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| 10 | |
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| 11 | <center> |
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| 12 | <h1> |
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| 13 | Sphlow Flotation Calculator Version 0.2</h1></center> |
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| 14 | |
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| 15 | <center><applet code="Sphlow.class" width=800 height=500 codebase=.>You |
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| 16 | are using a Java-challenged browser. Sorry!</applet><p> |
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| 17 | |
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| 18 | <a href="#targ0">About Sphlow</a> - |
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| 19 | <a href="#targ1">How to use it</a> - |
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| 20 | <a href="#targ2">How it works</a> - |
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| 21 | <a href="#targ3">Discussion</a> - |
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| 22 | <a href="#targ4">Known bugs</a> - |
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| 23 | <a href="#targ5">Credits and source</a></center> |
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| 24 | |
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| 25 | <h3><a NAME="targ0">About Sphlow:</h3> |
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| 26 | Sphlow was written to model metallurgical processes involving flotation of |
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| 27 | inclusions, such as tundishes or electron beam melting hearths. Inclusions |
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| 28 | are second-phase particles, such as oxide or nitride impurities, tungsten |
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| 29 | carbide tool bit chips left over from machining operations, etc. which |
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| 30 | get into metal and must be removed while the metal is molten. A given |
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| 31 | tundish or hearth design and operating parameter set will lead to a rough |
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| 32 | critical rising/sinking velocity for removal of all inclusions, such that |
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| 33 | (nearly) all inclusions rising or sinking faster than that critical velocity |
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| 34 | will be removed in that vessel. So, if inclusions are distributed |
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| 35 | in size and density (<i>D</i> and <i>rho<sub>p</sub></i>), then since velocity |
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| 36 | increases with diameter, those particles in the distribution above the |
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| 37 | critical velocity contour in the graph on the right will be removed from |
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| 38 | the molten metal by floating to the top or sinking to the bottom of the |
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| 39 | vessel. |
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| 40 | <p>Of course, you're free to use it for whatever purpose you like.</div> |
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| 41 | |
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| 42 | <h3><a NAME="targ1">How to use it:</h3> |
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| 43 | To draw the first plot, hit return in the velocity field. You may |
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| 44 | then use the menus and change the value in any field, just make certain |
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| 45 | you press <return> after each entry so that it will register. |
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| 46 | To zoom in on a graph, drag the mouse from the top left corner to the bottom |
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| 47 | right corner of what you'd like to see. To zoom out, drag the mouse |
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| 48 | upward. You can also change the limits in the <b>"D-rho curve controls"</b> |
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| 49 | panel. |
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| 50 | <p>So, type in your fluid viscosity and density, or choose from the liquids |
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| 51 | offered. When you give it a critical velocity, Sphlow will tell |
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| 52 | you the Reynolds number, friction factor and particle diameter and show |
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| 53 | you the relevant point for the given particle density in the friction factor-Reynolds |
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| 54 | number curve on the left. It will also plot diameters which rise/fall |
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| 55 | at the same velocity for a range of particle densities in the curve on |
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| 56 | the right. You may plot up to nine such velocity contours by selecting |
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| 57 | a new curve number and entering a new velocity. If you change liquid |
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| 58 | properties or <i>D-rho</i> graph limits, Sphlow will redraw all of |
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| 59 | the velocity contours. |
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| 60 | <p>The red vertical line on the right indicates the liquid density. |
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| 61 | The starting point corresponds to nickel spheres in water.</div> |
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| 62 | |
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| 63 | <h3><a NAME="targ2">How it works:</h3> |
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| 64 | This applet calculates and displays the relationship between size, density |
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| 65 | and terminal rising/sinking velocity of a spherical particle in a fluid. |
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| 66 | It is based on the relationship between friction factor and Reynolds number |
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| 67 | in laminar flow, from Stokes flow where <i>Re</i><0.2 to Newton's law |
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| 68 | flow for Reynolds numbers from about 10<sup>3</sup> to 10<sup>5</sup>. |
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| 69 | At each point, it first calculates the ratio <i>f/Re</i> (friction factor |
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| 70 | divided by Reynolds number) given by: |
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| 71 | <br> |
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| 72 | <center><table CELLSPACING=0 CELLPADDING=0 NOSAVE > |
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| 73 | <tr ALIGN=CENTER NOSAVE> |
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| 74 | <td ALIGN=CENTER NOSAVE><i><u>f</u></i></td> |
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| 75 | |
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| 76 | <td ALIGN=CENTER VALIGN=CENTER ROWSPAN="2" NOSAVE>=</td> |
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| 77 | |
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| 78 | <td NOSAVE><u>4 <i>mu g</i> |<i>rho<sub>p</sub></i>-<i>rho<sub>f</sub></i>|</u></td> |
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| 79 | </tr> |
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| 80 | |
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| 81 | <tr ALIGN=CENTER NOSAVE> |
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| 82 | <td ALIGN=CENTER NOSAVE><i>Re</i></td> |
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| 83 | |
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| 84 | <td><i>rho<sub>f</sub></i><sup>2</sup><i>u</i><sup>3</sup></td> |
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| 85 | </tr> |
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| 86 | </table></center> |
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| 87 | where <i>mu</i> and <i>g</i> are the viscosity and gravitational acceleration |
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| 88 | respectively, <i>rho<sub>f</sub></i> and <i>rho<sub>p</sub></i> the fluid |
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| 89 | and particle densities, and <i>u</i> the rising or sinking velocity. |
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| 90 | It then draws a diagonal line of slope 1 in the graph on the left corresponding |
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| 91 | to that value of <i>f/Re</i>, and the intersection with the red curve gives |
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| 92 | the Reynolds number and friction factor. Since |
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| 93 | <br> |
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| 94 | <center><table CELLSPACING=0 CELLPADDING=0 NOSAVE > |
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| 95 | <tr ALIGN=CENTER NOSAVE> |
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| 96 | <td ALIGN=CENTER VALIGN=CENTER ROWSPAN="2" NOSAVE><i>Re</i></td> |
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| 97 | |
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| 98 | <td ROWSPAN="2" NOSAVE>=</td> |
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| 99 | |
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| 100 | <td><i><u>rho<sub>f</sub> u D</u></i> ,</td> |
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| 101 | </tr> |
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| 102 | |
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| 103 | <tr ALIGN=CENTER VALIGN=CENTER NOSAVE> |
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| 104 | <td NOSAVE><i>mu</i></td> |
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| 105 | </tr> |
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| 106 | </table></center> |
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| 107 | the particle diameter <i>D</i> follows straightforwardly.</div> |
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| 108 | |
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| 109 | <h3><a NAME="targ3">Discussion:</h3> |
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| 110 | This model assumes a quiescent liquid. If there is a lot of vertical |
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| 111 | flow, recirculation or turbulence in your problem, particles will be swept |
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| 112 | up and down, so it is wise to be conservative in your estimate of critical |
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| 113 | velocity (<i>i.e.</i> use a <b>much</b> larger velocity than the vessel |
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| 114 | height divided by residence time). |
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| 115 | <p>The model also assumes constant liquid and particle densities. |
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| 116 | If these densities vary strongly with temperature, things will be somewhat |
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| 117 | more complicated: if the liquid coefficient of thermal expansion (CTE) |
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| 118 | is higher than that of the particle, there will be a range of densities |
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| 119 | at which particles will be neutrally buoyant; if the particle CTE is higher |
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| 120 | then it will always float or sink. Density also changes in porous |
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| 121 | inclusions, <i>e.g.</i> porous titanium nitride in molten titanium, where |
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| 122 | dissolving the dense nitride decreases a particle's average density but |
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| 123 | filling a pore increases it. (See the <a href="http://www.tms.org/pubs/journals/MT/B/9712/abstracts-9712.B.html#5">paper |
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| 124 | of Jean-Pierre Bellot and Alec Mitchell</a> in the <a href="http://www.tms.org/pubs/journals/MT/B/9712/contents-9712.B.html">12/1997 |
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| 125 | issue</a> of <i><a href="http://www.tms.org/pubs/journals/MT/MT.html">Metallurgical |
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| 126 | and Materials Transactions</a></i> for details. Hey, while you're |
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| 127 | there, check out <a href="http://www.tms.org/pubs/journals/MT/B/9712/abstracts-9712.B.html#29">my |
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| 128 | paper</a> too!) |
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| 129 | <p>Finally, the model assumes spherical inclusion particles. For |
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| 130 | non-spherical particles, the friction factor will usually be smaller than |
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| 131 | that of the smallest sphere containing the particle (always so in the case |
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| 132 | of Stokes flow where <i>Re</i><0.2), so velocity will be higher, and |
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| 133 | this model will give a conservative velocity estimate for such particles.</div> |
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| 134 | |
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| 135 | <h3><a NAME="targ4">Known bugs:</h3> |
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| 136 | Due to a bug in the ptplot package, the "fill" button will redraw the plot |
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| 137 | with limits set to encompass all points that have ever been drawn, not |
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| 138 | merely those which are currently active. It should not be necessary |
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| 139 | to use this button. Another bug in ptplot limits the minimum size |
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| 140 | of the graphs, so a version smaller than 800x500 will not be feasible anytime |
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| 141 | soon. Also, some of the velocity contours are not drawn to the top |
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| 142 | of the <i>D-rho</i> graph space; I will fix that at some point. |
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| 143 | <p>The javascript layers business seems to be broken on newer versions |
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| 144 | of Netscape. Oh well, I may try to fix it someday.</div> |
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| 145 | |
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| 146 | <h3><a NAME="targ5"><a href="http://www.jars.com/"><img SRC="jarsregistered.gif" ALT="JARS-registered" BORDER=0 height=93 width=141 align=RIGHT></a>Credits |
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| 147 | and source:</h3> |
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| 148 | Sphlow is based in large part on the <a href="http://ptolemy.eecs.berkeley.edu/java/ptplot/index.html">ptplot |
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| 149 | package</a> at UCBerkeley, copyright Regents of the University of California. |
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| 150 | I also learned a lot from the <a href="http://www.phrantic.com/scoop/onjava.html">Scoop |
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| 151 | on Java</a> (where my <a href="http://lyre.mit.edu/~powell/Software/Java/hello.html">first applet</a> came from), |
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| 152 | and the <a href="http://lyre.mit.edu/~powell/Software/Java/lemniscate.html">Lemniscate</a>, |
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| 153 | among the <a href="http://www-history.mcs.st-and.ac.uk/~history/Java/">curves</a> |
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| 154 | from the <a href="http://www.st-andrews.ac.uk/">University of St. Andrews</a>. |
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| 155 | Byte code for most of the classes is due to the SGI JDK 1.1 (DiamCalculator.class |
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| 156 | was recompiled in May, 2000 using guavac 1.2 to fix a bug), but it should |
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| 157 | all be JDK 1.0 compliant. |
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| 158 | The Magic Red Curve in the friction factor graph on the left is based on |
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| 159 | principles of fluid flow around a sphere, and taken from <i><a href="http://www.amazon.com/exec/obidos/ASIN/047107392X">Transport |
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| 160 | Phenomena</a></i> by Bird, Stewart and Lightfoot. |
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| 161 | |
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| 162 | <p>Get the <a href="./">source |
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| 163 | code here</a> (distributed under <a href="http://www.gnu.org/copyleft/gpl.html">GPL</a>). |
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| 164 | <p>Email compliments/complaints/comments, bug reports and suggestions for |
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| 165 | new liquids to <a href="mailto:Adam Powell <apowell@opennovation.com>">Adam |
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| 166 | Powell</a>.</div> |
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| 167 | |
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| 168 | </body> |
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| 169 | </html> |
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