Today, an engineering student voices a concern. The
University of Houston presents this series about
the machines that make our civilization run, and
the people whose ingenuity created them.

Michelle Ramotowski tells
how her teacher finishes off a proof with the
words, "Problem solved!" He sets the problem up,
draws a diagram -- then, while the class waits for
equations, he abruptly says, "Problem solved!" But
he's only uttered words. Mere logic, she tells us,
seems out place in this world of science and
engineering. How can a solution lie in words alone!

She asks us to look at cooling fins, for example --
the kind you see on radiators, at home or in your
car -- the kind that surround your overheated
computer chip. Those sheet-metal plates help a hot
object shed heat by increasing its surface area.

In 1926 the great German engineer Ernst Schmidt
wrote about fins. He argued that a certain optimal
shape would give the best cooling for a given
amount of metal. He wrote no equations. He simply
likened the flow of heat in a fin to the flow of
water in a pipe. His physical insight led him to a
simple parabolic cross-section. The argument was
one paragraph long.

During the Sputnik years, we engineers grew far
more formal and mathematical. By 1959 we were no
longer willing to hear Schmidt's simple reasoning;
and, in 1959, R.J. Duffin wrote:

*It seems that the literature does not contain a
proof of the Schmidt criterion. Schmidt did advance
an intuitive argument, but it is not
convincing.*

Duffin wrote six pages of the calculus of
variations. He finally convinced himself Schmidt'd
been right all along. So why had Schmidt felt no
need to turn mathematics on this problem? Well, in
1926 the logic of physical processes was not yet
out of style.

Ramotowski thinks Schmidt offers a lesson to us all
-- to her faculty and to her fellow students. "When
I try to explain technical ideas to non-engineers,"
she says, "my equations draw blank stares. When I
use pictures and analogies, the person understands.
We get so caught up in calculation that we forget
to step back and see the picture whole."

So she takes us back to the fin problem. In 1974,
C.J. Maday wrote a new mathematical model for the
optimal fin shape. He got the same overall form
Schmidt did, but with a wavy surface ending in a
tip whose shape was simply silly. "Sure, the proof
is beautiful," she says, "but the thing obviously
wouldn't work."

When you write mathematical models, you simplify
problems and narrow their scope. That's a good
thing to do until you give away major pieces of
reality -- until mathematics replaces mental fight
instead of joining it. The blind men trying to
describe an elephant learned a great deal by
looking at parts separately -- by writing equations
for the trunk or the tail. But, in the end, we have
to step back and see the elephant whole. Only when
we do, can we ever really say -- "Problem solved!"

I'm John Lienhard at the University of Houston,
where we're interested in the way inventive minds
work.

(Theme music)

Schmidt, E., Die Wärmeübertragung durch
Rippen. Zeitschrift des Vereines Deutscher
Ingenieure, Band 70, Nr. 28, 26 Juni 1926, S.
885-889.
Duffin, R. J., A Variational Problem Relating to
Cooling Fins. Journal of Mathematics and
Mechanics, Vol. 8, No. 1, 1959, pp. 47-56.

Maday, C.J., The Minimum Weight One-Dimensional
Straight Cooling Fin. Transactions of the
ASME. Journal of Engineering for Industry,
Vol. 96, No. 1, 1974, pp. 161-165.

For an introductory discussion of cooling fin
design, see: J. H. Lienhard IV and J. H. Lienhard V, A Heat Transfer Textbook, 5th ed., Dover Pubs. Inc., Mineola, NY, 2019. You can easily download the entire book, free of charge, at https://ahtt.mit.edu/ ,
Section 4.5.

This episode was conceived and drafted by Michelle
Ramotowski, a student in the Mechanical Engineering
Department at UH. N. Shamsundar, UH Mechanical
Engineering Department, contributed significant
counsel.

The Engines of Our Ingenuity is
Copyright © 1988-1997 by John H.
Lienhard.

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