Getting Lukewarmer

pCup

Last November, I was in my car on a Friday afternoon and listening to WBUR.  Regular listeners will know that Friday afternoons are special because it’s when Science Friday is broadcasted.  On that particular Friday, the topic was about how long mammals take to urinate.  Apparently, some researchers have shown that most mammals take about the same amount of time to urinate, regardless of their size.  It’s pretty amazing, if you think about it.  My mother’s cat (Ashby) takes as long to pee as the elephants in the Big Apple Circus.  If you’re curious, here’s a link to the Science Friday story, though it wasn’t the original bit that aired.

Back to my story.  So I’m chuckling to myself, imagining researchers with stopwatches timing various mammals during their bathroom breaks.  Science Friday breaks to an interview with the lead researcher, a professor at Georgia Tech.  Of course!  I know him!  Of anyone who would do research like this, it would be Dave Hu.  Who else?

Let me give you a bit of a back story.  I have known Dave since 1999.  At that time, we were both taking mechanical engineering courses.  I remember late nights at the student center cranking out problem sets with Dave and company.  It was always memorable because…  Well, if you met Dave, you would know why.  Dave is one of those people who speaks whatever is on his mind.  And whatever is on is mind is almost never what is on other people’s minds.

Eventually, Dave went on to get his Ph.D. in math.  In fact, we lived in the same dorm in graduate school.  He and another great friend of mine, Brian Chan, studied how water striders moved.  They made the cover of Nature in 2003.  That was what began a string of research topics in math and dynamics found in nature for both Dave and Brian.

Now back in my car on that Friday afternoon, my chuckle has now crescendo’ed into a full-blown “I can’t believe I’m hearing Dave Hu on NPR” laughter.  As soon as I got home, I e-mailed Dave to congratulate him.  I also asked him for a copy of the paper in which he discusses the details.  Unfortunately, the paper wasn’t published yet, so it wasn’t ready for public viewing.  But it’s finally here!

I know that scientific journal publications can sometimes be daunting, but I do think it’s worth a read.  It’s quite short, and Dave has a way of presenting things in a reader-friendly way.  If anything, just read the first few pages, through section 4.  To summarize the paper, bladders can range greatly between mammals.  But urethras among mammals also range greatly.  So while larger mammals have larger bladders to empty, they also have longer urethras, which means there is a greater gravitational force driving the flow.

In slightly more detail, Dave uses scaling laws to approximate the radius of a drop of urine and also the time to eject one drop.  Combining these two, he approximates the total time it takes to empty a bladder of a given size.  The results show that this time is a constant, indicating that mammals, whether  1kg or 100kg, take roughly the same amount of time to urinate.

Dave and his team filmed different animals urinating and also took measurements from videos from YouTube.  The animals they measured spanned 0.03kg to 8000kg.  The time to empty a bladder was T = 21 ± 13 seconds.  This, of course, has led me to thinking about my own bodily functions.  I haven’t exactly timed myself in the bathroom, but I’m pretty much in that range.  We all know from experience, the duration can vary greatly based on how full your bladder is.

As much as I am dedicated to math and science, I have not reached to point of subjecting my body to reach a full bladder’s worth of liquid before relieving myself.  But I’ve done the next best thing.  I have decided to apply the equation (equation 1 in the paper, dubbed the Law of Urination) for urination time to myself.  Ok, so not exactly myself, but humans in general.

I had to look up the average values for urethra length and diameter, bladder size, etc.  The bladder (or intravesical) pressure was a bit more difficult to find.  But this particular paper seems to suggest that the pressure is about 40cmH2O, which is about  4000Pa.

Law_of_Urination

Putting all these values into the equation, I get a time of 15 seconds.  A fun exercise would be to tinker with the numbers to get different tinkle times.  I leave the exercise of comparing the theoretical result with experimental data up to the reader!

(Addendum: A quick consult with my former roommate has led to some newer numbers.  Dr. Allen Chang cited this article, which lists 12mmHg (or 16.3cmH2O) as the upper limit of the intra-abdominal pressure.  The pressure in the bladder is in line with the pressure in the abdominal cavity, so intra-abdominal pressure is a good measure of bladder pressure.  The 40cmH2O number I had used earlier was for a fully distended bladder, which is a very extreme case.  Using 16.3cmH2O for bladder pressure gives us a new time of 19.5 seconds, just 1.5 seconds off from the 21 second “Golden Law.”)

Download the number one worksheet here!

About Roger Yeh @ PTC

Roger is a mechanical engineer by training. He studied non-Newtonian fluid mechanics (specifically the extensional flow of polymer solutions) in graduate school. Since graduating, he has dabbled in systems engineering (integrating hardware and software components on a radar program) and test and measurement applications (across all industries). He joined PTC in March 2011 as an application engineer in the Mathcad Business Unit.
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