Monday, October 4, 2010

It hasn't been that long since my last post

    It's been about 7 months, maybe 200 days or so, since my last post.  And while you may not have been counting the 4800 hours, you probably aware of the 5 million barrels or 210 million gallons of oil that were being dumped into the gulf.  This recent SEED slide show highlights the our deficiency as humans when it comes to comprehending these sorts of massive numbers.  Perhaps this is part of the reason we cannot stop buying drinks in disposable containers.  The images in the slide show come from Chris Jordan's 2009 book entitled Running The Numbers .  Of course, the idea of using technology and imagery to help us wrap our minds around the gargantuan nature of our world is nothing new.  The short video Powers of Ten from the 1960's highlights the wonders of the universe by expanding our field of view by one power of ten every 10 seconds.  In the video, we see a square that is 10^8 meters on a side framing the earth and a square that is 10^-8 meters per side framing a coil of DNA.

   Recently, while teaching my Math 124 Calculus Course, I came up with a little related rates program to help the students wrap their minds around the spreading of the oil in the gulf coast.  According to a NY times article from this summer, anywhere from 12 to 25 thousand barrels of oil per day were being dumped.  Wikipedia's site on oil slicks asserts that an oil slick is no thicker than about .002 millimeters.
There are 42 gallons of oil in a barrel.  So we will go with the round and reasonable number of about 1 million gallons per day.  A quick conversion gives us

(1 millions gallons/day) (3.79 liters/gallon) (10^6 mm^3/liter) = 3.79 (10^12) mm^3/day

Now we make a few fairly broad assumptions:

  1. The oil spreads in a circular manner and always has uniform thickness. 
  2. The oil is being spilled at a constant rate.
Most of us have experienced that as liquid spills out onto a surface, it spreads quickly at first and then slows down over time even if we do pour at a constant rate.  But how exactly is the rate of change of the radius of the spill related to the current radius of the spill?

We now answer that question with a little calculus.
Suppose V(t) denotes the volume of oil spilled at a time t where t is measured in days.  According to the assumptions we made,
V(t)=pi* r(t)^2 h where r(t) is the radius of the slick in millimeters at time t (in days) and h is the thickness of the slick.  So, using the chain rule, we have V'(t)= 2pi*h*r(t)*r'(t).
Because we know that the rate of change of volume of the oil spilled is constant at 3.79 (10^12) mm^3/day, we can determine the rate of change of the radius with respect to time in terms of the radius of the slick at a particular time.  So we see that r'(t)*r(t) is approximately 302(10^12)mm^2/day.  If we choose to measure r(t) in kilometers, then r'(t)*r(t) is approximately 302 km^2/day.
In other words, the rate of change of the radius of the slick is inversely proportional to the current radius.
So, when the slick is 1 km in radius, it is spreading at a rate of 302 kilometers per day.
But when the radius is 302 km, the radius is changing at a rate of only 1 kilometer per day.
By the way, 302 km is about 188 miles.  But the rate of spread is slowing, so how long would it take for the oil to reach shore if the spill occurred 100 miles off shore?  If the rate of the spill really does stay constant (i.e. no successful clean up) then we see can approximate r(t) as 25 t^(1/2).   When is 100=25t^(1/2)?  After about 16 days. So a major oil spill, even if it occurred 100 miles out into the ocean could reach the shore in a few weeks!
   I can only hope that the spread of information is as successful and uniform.  And I also make the observation that I waited about the same number of days to post as the number of kilometers that the oil slick would have grown in even one day.

P.S. How accurate is this little estimate?  Certainly we made many simplifications and overlooked all clean up efforts.  Looking at an interesting app from the NY times, I see that it did indeed take about three weeks for the spill to be noticed on the shore of Mississippi about 100 miles away from the source.

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