Analytic Solution for the Cylinder

<BODY LANG="EN"> <HR><center> <TR> <table border=1> <td><A HREF="node9.shtml"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A HREF="node9.shtml"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A HREF="node11.shtml"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node10.html"><img height=36 width=36 alt="CONTENTS" src="/UCESIcons/tools/toc.gif"></a></td> <td><a href="reset.hamlet"><img height=36 width=36 alt="RESET" src="/UCESIcons/tools/stop.gif"></a></td> <td><a href="copyright.html"><img height=36 width=36 alt="COPYRIGHT" src="/UCESIcons/tools/copyright.gif"></a></td> <td><a href="mailto:uces_info@krellinst.org"><img height=36 width=36 alt="FEEDBACK" src="/UCESIcons/tools/feedback.gif"></a></td> <td><a href="http://www.cs.utah.edu/%7Ehamlet/icons.html"><img height=36 width=36 alt="ABOUT" src="/UCESIcons/tools/about.gif"></a></td> </TR> </table> </center> <HR> <!DOCTYPE HTML PUBLIC "-//IETF//DTD HTML 2.2//EN">  <HTML> <HEAD> <TITLE>Analytic Solution for the Cylinder</TITLE> <BASE TARGET="_parent"> <META NAME="description" CONTENT="Analytic Solution for the Cylinder"> <META NAME="keywords" CONTENT="cone"> <META NAME="resource-type" CONTENT="document"> <META NAME="distribution" CONTENT="global"> <LINK REL=STYLESHEET HREF="cone.css"> </HEAD> <BODY LANG="EN"> <H2>Analytic Solution for the Cylinder</H2> <P> Although it may not be practical to solve most vessel drainage problems analytically, we can learn a lot about the behavior of such systems in general by examining the solution of a simple case. By using a computer algebra software package, such as <nobr>M<font size=-1><small>APLE</small></font></nobr> or <nobr>M<font size=-1><small>ATHEMATICA</small></font>.</nobr> <P> Rearranging slightly Equation (17) from above, we have our mathematical model for the draining cylinder in a form suitable for submitting to M<font size=-1><small>APLE</small></font> for solution. <P> <P> <IMG WIDTH=500 HEIGHT=45 ALIGN=BOTTOM ALT="[Eq. 19]" SRC="img48.gif"> <P> where we have defined <IMG WIDTH=16 HEIGHT=22 ALIGN=MIDDLE ALT="R_c" SRC="img27.gif"> to be the radius of the cylinder whose volume will be <EM>exactly</EM> the same as that of the cone when the initial liquid level is <IMG WIDTH=13 HEIGHT=14 ALIGN=MIDDLE ALT="y_0" SRC="img37.gif"> when measured from the drain orifice. <P> <B>Exercise:</B> <EM>Verify that under these conditions the required cylinder radius is </EM> <P> <IMG WIDTH=349 HEIGHT=33 ALIGN=BOTTOM ALT="[eq. for R_c" SRC="img49.gif"> <P> We assume that you are familiar with the basic operations in <nobr>M<font size=-1><small>APLE</small></font>. </nobr> You will find that it is frequently a good idea to initialize your variables and parameters in <nobr>M<font size=-1><small>APLE</small></font>,</nobr> since once numerical values are stored in these symbols, they remain there. That means, that if you want to start your calculations over, you need to re-initialize them. You'll see what we mean as you gain experience. <P> Below is a typical statement for initializing the <nobr>M<font size=-1><small>APLE</small></font></nobr> variables we will use: The <nobr>M<font size=-1><small>APLE</small></font></nobr> commands are: <P> <a href="sende1.hamlet"><img height=24 width=24 alt="EXECUTE" src="/UCESIcons/tools/execute.gif"></a> <a href="showe1.html" TARGET="results"><img height=24 width=24 alt="SHOWCODE" src="/UCESIcons/tools/show_code.gif"></a> <P> If you like, you can simply use your mouse to `swipe' the commands in the browser window, then paste them in the <nobr>M<font size=-1><small>APLE</small></font></nobr> window. Then press the <I>enter</I> key. <P> Now to submit the equation we wish to solve to <nobr>M<font size=-1><small>APLE</small></font>,</nobr> we define, for convenience, a variable called <I>de</I>1 in which to store the equation. The necessary <nobr>M<font size=-1><small>APLE</small></font></nobr> command is <P> <a href="sende2.hamlet"><img height=24 width=24 alt="EXECUTE" src="/UCESIcons/tools/execute.gif"></a> <a href="showe2.html" TARGET="results"><img height=24 width=24 alt="SHOWCODE" src="/UCESIcons/tools/show_code.gif"></a> <P> Next we turn this <EM>ordinary differential equation</EM> over to the differential equation solver in <nobr>M<font size=-1><small>APLE</small></font>,</nobr> called <B>dsolve</B>. We will will define a variable <I>y</I>1(<I>t</I>) to be the solution of <I>de</I>1. <P> <a href="sende3.hamlet"><img height=24 width=24 alt="EXECUTE" src="/UCESIcons/tools/execute.gif"></a> <a href="showe3.html" TARGET="results"><img height=24 width=24 alt="SHOWCODE" src="/UCESIcons/tools/show_code.gif"></a> <P> The solution is not very elegant looking. In fact, it's virtually impossible to tell anything about the behavior of this physical system by looking at the solution to the mathematical model. If a picture is worth a thousand words, then a plot of the solution is worth about the same. So let's choose some numerical values and plot the results to see what the solution <EM>looks</EM> like. <P> First, let's tell <nobr>M<font size=-1><small>APLE</small></font></nobr> what <IMG WIDTH=16 HEIGHT=22 ALIGN=MIDDLE ALT="R_c" SRC="img27.gif"> is in terms of variables we will use later in the calculations involving the cones. <P> <a href="sende4.hamlet"><img height=24 width=24 alt="EXECUTE" src="/UCESIcons/tools/execute.gif"></a> <a href="showe4.html" TARGET="results"><img height=24 width=24 alt="SHOWCODE" src="/UCESIcons/tools/show_code.gif"></a> <P> Then let us choose some reasonable physical values in the <I>SI</I> system of units, with all lengths in <I>meters</I> and <I>g</I> in <I>meters</I>/<I>sec</I><sup><small>2 </small></sup>.  <P> <a href="sende5.hamlet"><img height=24 width=24 alt="EXECUTE" src="/UCESIcons/tools/execute.gif"></a> <a href="showe5.html" TARGET="results"><img height=24 width=24 alt="SHOWCODE" src="/UCESIcons/tools/show_code.gif"></a> <P> Note carefully that we have made certain that the radius of the drain <IMG WIDTH=12 HEIGHT=14 ALIGN=MIDDLE ALT="r_e" SRC="img12.gif"> is smaller than the minor radius of the cone <I>r</I>. Now to produce a plot let's call on <nobr>M<font size=-1><small>APLE</small></font></nobr> again. Since we have no prior knowledge of how long the cylinder might take to drain (in seconds), we might try solving algebraically for <I>t</I> in our solution from above. But we see right away that <nobr><I>y</I>(<I>t</I>)=<I>y</I>1(<I>t</I>)</nobr> is quadratic in <I>t</I> in a very messy way. However, if we examine our solution in light of the fact that <IMG WIDTH=52 HEIGHT=22 ALIGN=MIDDLE ALT="R_c > r_e" SRC="img51.gif"> so that <IMG WIDTH=67 HEIGHT=27 ALIGN=MIDDLE ALT="R_c^4 >> r_e^4" SRC="img52.gif"> , then we will find that the time for the cylinder to empty completely will be about 30 minutes, or 1760 seconds. <P> <B>Exercise:</B> <EM>Go back to Equation (17) and eliminate the radical in the denominator, </EM> <P> <IMG WIDTH=284 HEIGHT=50 ALIGN=BOTTOM ALT="[sqrt expression]" SRC="img53.gif"> <P> <EM>and obtain the solution to this equation using <nobr>M<font size=-1><small>APLE</small></font>.</nobr> (This model neglects the velocity of the free surface of the liquid in computing the exit velocity. That is, it assumes <IMG WIDTH=116 HEIGHT=31 ALIGN=MIDDLE ALT="[expression for v_exit]" SRC="img54.gif"> .) Now use the resulting solution to estimate the drainage time by solving for <I>t</I> when <nobr><I>y</I>(<I>t</I>)=0.</nobr> You can do this using the <B>solve</B> and <B>evalf</B> commands in <nobr>M<font size=-1><small>APLE</small></font>.</nobr> <P> To produce the plot of <IMG WIDTH=13 HEIGHT=14 ALIGN=MIDDLE ALT="y_1" SRC="img55.gif"> <I>vs t</I>, the appropriate <nobr>M<font size=-1><small>APLE</small></font></nobr> statement is:</EM> <P> <a href="sende6.hamlet"><img height=24 width=24 alt="EXECUTE" src="/UCESIcons/tools/execute.gif"></a> <a href="showe6.html" TARGET="results"><img height=24 width=24 alt="SHOWCODE" src="/UCESIcons/tools/show_code.gif"></a> <P> </BODY> </HTML> <HR><center> <table border=1> <TR> <td><A HREF="node9.shtml"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A HREF="node9.shtml"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A HREF="node11.shtml"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node10.html"><img height=36 width=36 alt="CONTENTS" src="/UCESIcons/tools/toc.gif"></a></td> <td><a href="reset.hamlet"><img height=36 width=36 alt="RESET" src="/UCESIcons/tools/stop.gif"></a></td> <td><a href="copyright.html"><img height=36 width=36 alt="COPYRIGHT" src="/UCESIcons/tools/copyright.gif"></a></td> <td><a href="mailto:uces_info@krellinst.org"><img height=36 width=36 alt="FEEDBACK" src="/UCESIcons/tools/feedback.gif"></a></td> <td><a href="http://www.cs.utah.edu/%7Ehamlet/icons.html"><img height=36 width=36 alt="ABOUT" src="/UCESIcons/tools/about.gif"></a></td> </TR> </table> </center> <HR> <P> <ADDRESS> <I>Boyd Gatlin<BR> Associate Professor of Aerospace Engineering<BR> Mississippi State University</I>  </ADDRESS> </BODY>