Nusselt, Biot numbers and Ozisik

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The non-dimensional representation of the convection heat transfer coefficient ‘h’ is identified as the Nusselt number, in honor of Wilhelm Nusselt. It can be written as

$Nu = \frac{hL}{k} \cdots (1)$

In Eq. (1), L is a characteristic length scale. For instance, if one needs to define the overall convection heat transfer coefficient for a (cold) flow over a (hot) flat plate, then L would represent the total finite length of the flat plate along the flow direction. The ‘k’ in Eq. (1) is the thermal conductivity of… let us wait and proceed.

There is another non-dimensional number in heat transfer physics called the Biot number, named after Jean-Baptiste Biot (whose other contributions include the Biot-Savart law and according to some, the Fourier Law). It is written as

$Bi = \frac{hL}{k} \cdots (2)$

where h is the convection heat transfer coefficient, L is the characteristic length scale, ‘k’ is the thermal conductivity of… wait a minute, aren’t Eq. (2) and Eq. (1) same?

In Eq. (2) the thermal conductivity ‘k’ is that of the solid medium which is dipped in a fluid. In Eq. (1) the thermal conductivity ‘k’ is that of the fluid medium.

Nusselt number, through the non-dimensionalization of the heat transfer coefficient in Eq. (1), quantifies how much the convection heat transfer could be higher when compared with the conduction heat transfer, if the fluid were stationary.

The Biot number in Eq. (2) provides a way to compare the conduction resistance within a solid body to the convection resistance external to that body (offered by the surrounding fluid) for heat transfer.

Say an hot steel rod of diameter L is quenched by dipping into stationary air. Since the convection coefficient for stationary air at the maximum is around $10 W/m^2K$ and the thermal conductivity of hot steel ranges between $50 < k < 25 W/mK$ (decreases with increase in T), the Biot number in Eq. (2) would be $Bi \ll 1$ provided the L is sufficiently small. This allows one to simplify the transient conduction heat transfer process within the steel rod by treating it as a lumped medium with a single temperature (practically no temperature difference from the center to the edge of the rod in radial direction) changing in time.

For thermal insulators (k is very small) kept in a strong convection situation, irrespective of the smallness of L, $Bi \gg 1$ could prevail, where the conduction inside the insulator would result in a spatio-temporal temperature difference, which can neither be neglected nor allow one to ‘lump’ the insulator with a single representative temperature.

The Bi provides a way to use proper method of analysis for appropriate situations.

We stop here to take diversion.

Undergrads taking first course in Heat Transfer on many instances don’t appreciate the above difference between Nu and Bi. This is a standard googly question in any oral exam that involves testing of heat transfer basics. To set the record straight, I wasn’t aware of the difference in my undergrad either.

The first time I read and understood about the difference between Nu and Bi was in my masters from the excellent book Heat Transfer, a basic approach by M. N. Ozisik. I distinctly remember the elation of this ‘secret knowledge’ and the impulsive rush propelling me through the hostel corridors to my friend’s room to test his ‘ignorance’ against mine that is now slightly reduced. He later suggested to keep the book ‘hidden’ with me throughout the semester (the library had only one copy of the book) to ensure others remain protected from such treasures.

We stop this story of how I ‘lost’ the book and paid the hefty library fine acceding to my conniving spirit, to take one more diversion.

My entire note above is just a preamble to mention this: Professor Emeritus M. Necati Ozisik (1923-2008) passed away in Oct 2008. An obituary has appeared in the latest issue of the International Journal of Heat and Mass Transfer. An highlight from the obituary text:

[…] Ozisik dedicated his life to education and research in heat transfer.

[…] He published more than three hundred research papers in international journals and conferences. He was the author of eleven books, most of them best-sellers that were re-edited several times and published in different languages. His personal characteristics were apparent in all these books, where the material was rigorously presented in a clear, organized and systematic manner. As a result, his books became standards in graduate and undergraduate courses in many countries. His main contributions included analytical, numerical and hybrid solution techniques for direct and inverse problems, for coupled and uncoupled heat transfer modes.

Thank you Prof. M. Necati Ozisik for making me appreciate the nuances of heat transfer at the right age.

Reference

[1] Professor Emeritus M. Necati Ozisik 1923-2008 [doi:10.1016/j.ijheatmasstransfer.2008.12.022]