The Lewis Number

Last week in this blog I discussed why dolphins don’t breathe through gills like fish do. The take-home message was that using gills would cool the blood to the temperature of the surrounding water, and reheating the blood to the temperature of the dolphin’s body would require a prohibitive amount of energy.

You might be wondering: is there some tricky way that we can adjust things so that oxygen can diffuse without a significant heat transfer? Perhaps alter how long the blood is in thermal and diffusive contact with the seawater so there is time for oxygen diffusion but not time for thermal diffusion. Might that save the day?

You can compare the mechanisms of molecular and thermal heat transfer using the Lewis number, which is a ratio of the molecular diffusion constant and the thermal diffusion constant. Russ Hobbie and I discuss the Lewis number in Problem 20 of Chapter 4 of Intermediate Physics for Medicine and Biology. For oxygen diffusing in water, the diffusion constant is about 2 x 10⁻⁹ m²/s. The diffusion constant for heat is equal to the thermal conductivity divided by the product of the specific heat capacity and the density, which for water is about 1.5 x 10⁻⁷ m²/s. Thus, the diffusion constant for oxygen is about one hundred times less than the diffusion constant for heat. In other words, heat diffuses one hundred times more readily than oxygen, so it’s difficult to imagine how you could ever devise a situation where you could transfer oxygen without transferring heat too. As we concluded last week, physics constrains biology.

If you are exchanging heat and oxygen in air the situation is a bit better: for air the diffusion constant of oxygen and of heat are roughly the same. You can’t have oxygen diffusion without heat diffusion, but at least you aren’t down by a factor of one hundred.

The Lewis number is one of those useful dimensionless numbers—like the Reynolds number and the Peclet number—that summarizes the relative importance of two physical mechanisms. Because these numbers are dimensionless, their values does not depend on the system of units you use.

This all sounds fine and good, so imagine my surprise when one of my Biological Physics students working on this week’s homework assignment told me that the definition of the Lewis number in IPMB differs from the definition used by other sources. Yikes! The question comes down to this: is the Lewis number defined as the molecular diffusion constant over the thermal diffusion constant, or as the thermal diffusion constant over the molecular diffusion constant? In one sense it does not matter which definition you use. Either definition will tell you that in water oxygen has a harder time diffusing than heat. The only difference is that in one case the Lewis number is 1/100, and in the other case it is 100. The definition is arbitrary, like which direction you call right and which you call left. Had the first person to talk about those two directions called right left and left right, it would make no difference; they are just labels. However, I concede that if everyone uses different labels, confusion results. If half the people call left left and the other half call left right, then giving directions will be difficult—you would have to verify that you used the same definition of left and right before you could tell someone how to get across town.

I decided to check that fount of all knowledge: Wikipedia (how did I grow up without it?). There the definition is the opposite of that in IPMB—“Lewis number (Le) is a dimensionless number defined as the ratio of thermal diffusivity to mass diffusivity.” The free dictionary says “A dimensionless number used in studies of combined heat and mass transfer, equal to the thermal diffusivity divided by the diffusion coefficient” and thermopedia says the same, as does this publication. In the book Air and Water, Mark Denny uses our definition, molecules over heat (perhaps I should say we use Denny’s definition, because I am pretty sure we used Air and Water as our source). Interestingly, the CRC Handbook of Chemistry and Physics (I looked at the 59th edition, which is the one sitting in my office) says heat over molecules, but then adds “N.B.: Lewis number is sometimes defined as reciprocal of this quantity”). My conclusion is that the definition is a bit uncertain, but Russ and I (and Denny) appear to have adopted the minority view. What should I do? I’ve added to the IPMB errata the following entry:

Page 109: At the end of Problem 20, add the sentence “Warning: the Lewis number is sometimes defined as the reciprocal of the definition used here.”