Wednesday, June 2, 2010

The Loss Term

Everybody else is blogging about the BP oil spill, so I thought I'd join the fray.  But from an engineering perspective.

As the annals of history evaluate and scrutinize the events in gory detail of case study after case study, I believe that adding the dispersant will emerge as the worst single decision in the entire crisis.

This problem is a lot like reactor theory.  You have a diffuse gas (oil) in a dispersive medium (seawater).  There is a source term (the leaking pipe).  The approximations of reactor theory were built for this kind of stuff.

The fundamental equation here is the scalar Helmholtz equation, drilled into every nuclear engineer's early education:

D is the diffusion coefficient; it's how fast the oil "bleeds" through the water.
Sigma(a) is the absorption coefficient.  This is the "loss term" that the title of the blog talks about.
S is the source term, and it's estimated at 12,000 barrels per day.

So, the "D del-squared phi" term describes how fast and how far the oil spreads.  S is the 12,000 barrels or 500,000 gallons per day input into out system.

The "sigma-a phi" term describes how fast it's absorbed by something in the water -- the loss term. The loss term is either:
1.  Falling out of solution and ending up on the seafloor,
2.  Washing up on beaches, or
3.  Being eaten by tiny microbes.

But those microbes take a long time to eat up oil.  This webpage and video by the makers of Corexit is helpful and informative.

As of May 25th (which is a week ago at this writing), over 830,000 gallons of Corexit had been dumped into the water to break up the large globules of oil into tiny droplets that dispersed.  In the equation above, this makes D go from a small number to a huge number, and spreads the oil farther.  Without dispersant, you'd have a large, globular, sticky mess.  With Corexit, you have an even larger mess that can distribute and diffuse over a much larger area.  It does not speed up the oil-eating process fast enough to allow the microbes to eat the oil before it washes ashore!

Let me say it again, in a different way: since there is way more oil being dumped than the microbes can ever hope to eat, the only effective loss term is basically washing up on shore.  The use of a dispersant is inappropriate in a problem this large.  Spreading Corexit only makes it worse, because the diffusion coefficient (D) is now huge, and the oil slick can spread over a much wider area.

Ah, the law of unintended consequences.  I think for oil spills where it's already contained, the use of dispersants make sense.  In this case, though, the dispersant allows the oil to spread over a larger area, and makes the problem worse.

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