Conservation of Mass

<BODY LANG="EN"> <HR><center> <table border=1> <TR> <td><A HREF="node3.shtml"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A HREF="node2.shtml"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A HREF="node5.shtml"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node4.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>Conservation of Mass</TITLE> <BASE TARGET="_parent"> <META NAME="description" CONTENT="Conservation of Mass"> <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>Conservation of Mass</H2> <P> What we need now is a model of the <EM>instantaneous rate of flow</EM> of fluid from the container, which we will now obtain by applying the principle of mass conservation. <P> An important assumption we are making is that the fluid velocity is the same all across the cross sectional area of the drain orifice, <nobr><IMG WIDTH=16 HEIGHT=22 ALIGN=MIDDLE ALT="A_b" SRC="img9.gif"> .</nobr> Of course, it is not, since friction between the liquid and the drain will cause the velocity to be much lower near the edges than it is at the center of the drain. However, if we take <IMG WIDTH=13 HEIGHT=14 ALIGN=MIDDLE ALT="v_b" SRC="img10.gif"> to be the spatial <EM>average</EM> velocity across the drain orifice, then the <EM>mass flow rate</EM> through it is given by <P> <IMG WIDTH=500 HEIGHT=18 ALIGN=BOTTOM ALT="[Eq. 5]" SRC="img11.gif"> <P> where <IMG WIDTH=16 HEIGHT=22 ALIGN=MIDDLE ALT="A_b" SRC="img9.gif"> is the cross-sectional area of the drain, taken to be a circle of constant radius <nobr><IMG WIDTH=12 HEIGHT=14 ALIGN=MIDDLE ALT="r_e" SRC="img12.gif"> .</nobr> This relationship for fluids of uniform density is given in any basic textbook on fluid mechanics. Typical units of mass flow rate are <nobr><I>kg</I>/<I>s</I></nobr> or <nobr><I>slugs</I>/<I>s</I>.</nobr> <P> Further, the <EM>volume flow rate</EM>, usually designated <I>Q</I>, is simply the mass flow rate divided by the constant density, <nobr><IMG WIDTH=7 HEIGHT=14 ALIGN=MIDDLE ALT="rho" SRC="img2.gif"> .</nobr> That is, <P> <IMG WIDTH=315 HEIGHT=18 ALIGN=BOTTOM ALT="[Q eqn.]" SRC="img13.gif"> <P> where typical units are  <nobr><I>m</I><sup><small>3</small></sup>/<I>s</I> ,</nobr> <nobr><I>ft</I><sup><small>3</small></sup>/<I>s</I> ,</nobr> or <nobr><I>gal</I>/<I>min</I>.</nobr> The assumption that the velocity given by Bernoulli's equation is the <EM>average</EM> velocity across the orifice is a bit bold and will give us an overly optimistic answer for drainage times. In fact, Bernoulli's equation will give us the <EM>maximum</EM> possible, or <EM>ideal</EM>, velocity. We could correct this idealized flow rate <I>Q</I> by conducting some experiments in which we measured fluid flow rates through the drain hole and then computed a <EM>discharge coefficient</EM>, <IMG WIDTH=17 HEIGHT=22 ALIGN=MIDDLE ALT="C_d" SRC="img16.gif"> , for the drain, where <IMG WIDTH=171 HEIGHT=24 ALIGN=MIDDLE ALT="C_d = Q_real/Q_ideal < 1.0" SRC="img17.gif"> . Then, our <EM>corrected</EM> flow rate would be given by <IMG WIDTH=171 HEIGHT=31 ALIGN=MIDDLE ALT="[corrected Q eq.]" SRC="img18.gif"> . But, since we are interested in merely comparing flow rates for identical drain orifices, the discharge coefficients would be the same anyway. So, for simplicity, we will just use <IMG WIDTH=46 HEIGHT=22 ALIGN=MIDDLE ALT="C_d = 1" SRC="img19.gif"> . <P> The question now arises as to what the velocity at the surface of the liquid is. That is, what is <nobr><IMG WIDTH=13 HEIGHT=14 ALIGN=MIDDLE ALT="v_a" SRC="img20.gif"> ?</nobr> It is tempting to set <IMG WIDTH=13 HEIGHT=14 ALIGN=MIDDLE ALT="v_a" SRC="img20.gif"> to zero, since it would seem to be small compared to the outflow velocity. (A look at the equations above reveals that for a given flow rate, the velocity is inversely related to the cross sectional flow area.) However, we can apply <EM>conservation of mass</EM> to get a rational approximation for the free surface velocity. Since the liquid is conserved, the flow rate through a circular area just below the free surface at any instant must be equal to the flow rate through the drain. That is, whatever volume of fluid leaves the top surface of the fluid must pass through the drain orifice. Mathematically, this can be stated as:<P> <IMG WIDTH=280 HEIGHT=14 ALIGN=BOTTOM ALT="Q_a = Q_b" SRC="img21.gif"> <P> <IMG WIDTH=295 HEIGHT=13 ALIGN=BOTTOM ALT="v_a A_a = v_b A_b" SRC="img22.gif"> <P> <IMG WIDTH=500 HEIGHT=18 ALIGN=BOTTOM ALT="[Eq. 6]" SRC="img23.gif"> <P> Combining Equation (6) (conservation of mass) with Equation (4) (conservation of mechanical energy) leads to <P> <IMG WIDTH=500 HEIGHT=45 ALIGN=BOTTOM ALT="[Eq. 7]" SRC="img24.gif"> <P> where we have replaced <IMG WIDTH=13 HEIGHT=14 ALIGN=MIDDLE ALT="y_a" SRC="img25.gif"> with <nobr><I>y</I>(<I>t</I>)</nobr> to emphasize that the point on the free surface will be in motion (<EM>i.e.</EM>, falling) as the liquid level falls. Thus <I>y</I> is a function of time. <P> The volume flow rate, which we now see cannot be constant since <IMG WIDTH=13 HEIGHT=14 ALIGN=MIDDLE ALT="v_b" SRC="img10.gif"> depends on <nobr><I>y</I>(<I>t</I>),</nobr> is then given by <P> <IMG WIDTH=500 HEIGHT=45 ALIGN=BOTTOM ALT="[Eq. 8]" SRC="img26.gif"> <P> Take note of the parameter <I>R</I>, which by our development is the radius of the vessel at the free surface of the liquid. For a simple cylinder, this is a constant,  <nobr><I>R</I><sub><small>c </small></sub>.</nobr> But for the cone, <I>R</I> becomes a function of the liquid level <nobr><I>y</I>(<I>t</I>),</nobr> so that <nobr><I>R</I>=<I>R</I>(<I>y</I>(<I>t</I>)).</nobr> A more general mathematical model of the flow rate is then <br><br> <IMG WIDTH=500 HEIGHT=45 ALIGN=BOTTOM ALT="[Eq. 9]" SRC="img28.gif"> </p> <p> So far in developing our model we have, without saying so, used what is called <i>control volume analysis</i>. A control volume is simply a region in physical space surrounded by a <i>control surface</i>, which we are free to draw any way we choose. Once we have selected a control volume, we can then apply conservation of mass to it; that is, we sum the rate of inflow and rate of outflow and set the difference equal to the rate at which the amount of mass inside the control volume is changing. This may seem embarrassingly obvious, but when stated in a mathematical model it becomes a powerful tool of analysis. In our analysis above, we chose a control surface that exactly corresponded with the boundaries of the container everywhere except at the free surface. There, we let the control surface lie some small distance below the free surface at an instant in time. With the control volume thus defined, the total amount of mass remained constant, and we were able to deduce that, <i>at any instant</i>, inflow=outflow and thus derive an expression for the outflow velocity.</p> </BODY> </HTML> <HR><center> <table border=1> <TR> <td><A HREF="node3.shtml"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A HREF="node2.shtml"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A HREF="node5.shtml"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node4.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>