The Rate of Volume Change

<BODY LANG="EN"> <HR><center> <table border=1> <TR> <td><A HREF="node5.shtml"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A HREF="node5.shtml"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A HREF="node7.shtml"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node6.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>The Rate of Volume Change</TITLE> <BASE TARGET="_parent"> <META NAME="description" CONTENT="The Rate of Volume Change"> <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"> <H3>The Rate of Volume Change</H3> <P> Equation (10) is clearly the mathematical model we have been seeking. It relates the change in the amount of fluid in the tank to the rate at which it flows out. This may seem obvious, and it is for this simple case. But for more complicated situations, where there may be multiple inflows and outflows, a systematic approach to the analysis is essential. Here, by substituting Equation (9) into it, we get <P> <IMG WIDTH=500 HEIGHT=45 ALIGN=BOTTOM ALT="[Eq. 11]" SRC="img32.gif"> <P> However, we know from elementary geometry that the volume occupied by the fluid in the container, <I>V</I>, is related to the vessel radius <nobr><I>R</I>(<I>y</I>)</nobr> and to the fluid height <nobr><I>y</I>(<I>t</I>).</nobr> Clearly, this relationship will be different for each vessel to be considered. <P> The volume of a simple cylinder or cone can be expressed in terms of a radius  <nobr><I>R</I><sub><small>c</small></sub></nobr> and a length, here just <I>y</I> measured from the bottom. That is, <P> <IMG WIDTH=318 HEIGHT=18 ALIGN=BOTTOM ALT="[Eq. for V_cylinder]" SRC="img33.gif"> <P> For a simple cone, the volume is given by <P> <IMG WIDTH=346 HEIGHT=33 ALIGN=BOTTOM ALT="[Eq. for V_{simple cone}]" SRC="img34.gif"> <P> where we now denote that <I>y</I> is a function of time, since the free surface will drop after the fluid begins draining. In this problem, however, we are not dealing with a simple circular cone, but with a <EM>frustum</EM> of a cone; that is, a cone whose pointed end has been chopped off. For a conical tank with the small end down, the tank must be truncated there in order to have a drain orifice. As can be verified with some algebra (or by looking in a handbook), the volume of a frustum of a cone is given by <P> <IMG WIDTH=380 HEIGHT=33 ALIGN=BOTTOM ALT="[Eq. for V_cone]" SRC="img35.gif"> <P> Here <I>r</I> is the (constant) smaller radius and <nobr><I>R</I>(<I>y</I>)</nobr> is the larger radius, which clearly changes as the liquid level changes. Earlier we introduced the radius of the drain, <IMG WIDTH=12 HEIGHT=14 ALIGN=MIDDLE ALT="r_e" SRC="img12.gif"> . This is <EM>not</EM> the same parameter as the radius <I>r</I> of the cone frustum at the small end. <P> The <EM>rate</EM> at which the volume of liquid changes is just the time derivative of the volume, <nobr><I>dV</I>/<I>dt</I>.</nobr> But note that since <I>y</I> is a function of time, for the cone the radius <nobr><I>R</I>(<I>y</I>)</nobr> is also a function of time. Defining <IMG WIDTH=17 HEIGHT=22 ALIGN=MIDDLE ALT="R_0" SRC="img36.gif"> to be the cone radius at <IMG WIDTH=13 HEIGHT=14 ALIGN=MIDDLE ALT="y_0" SRC="img37.gif"> , the radius of the cone varies as: <P> <IMG WIDTH=338 HEIGHT=38 ALIGN=BOTTOM ALT="[Eq. for R(t)]" SRC="img38.gif"> <P> When this expression is substituted into the one for <IMG WIDTH=35 HEIGHT=22 ALIGN=MIDDLE ALT="V_cone" SRC="img39.gif"> and the result differentiated with respect to time, we get the time rate of change of volume of fluid in the conical tank: <P> <IMG WIDTH=500 HEIGHT=40 ALIGN=BOTTOM ALT="[Eq. 12]" SRC="img40.gif"> <P> Since the large radius <I>R</I> is constant for the cylinder, the volume change rate is just the rate of change of <nobr><I>y</I>(<I>t</I>):</nobr> <P> <IMG WIDTH=500 HEIGHT=34 ALIGN=BOTTOM ALT="[Eq. 13]" SRC="img41.gif"> <P> You should verify these developments for yourself. <P> If we wished to know rate of change of <EM>mass</EM> inside the cylinder, we need only multiply the rates for volume by the density, which we specified to be a constant.<P> </BODY> </HTML> <HR><center> <table border=1> <TR> <td><A HREF="node5.shtml"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A HREF="node5.shtml"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A HREF="node7.shtml"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node6.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>