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coefficient of expansion figures

updated thu 12 sep 02


Roger Graham on wed 11 sep 02

Further to the recent postings on coefficient of expansion "COE" figures,
and the relation between coefficients of LINEAR expansion and CUBIC

First a bouquet for David Hewitt, and his wonderful web pages on calculating
the COE for a glaze. He's been there, done that. Thanks David. Go there
first and read it all, before we continue. Explore the murky world of the
confusing ways potters have described expansion figures in the past. The URL

Iandol (Hi, Ivor) raised the question whether the coefficient of LINEAR
expansion is simply one third of the CUBIC value, or whether it should
really be the cube root of the cubic value. No, the cube root idea doesn't
work, because these "coefficient" values refer to the CHANGE in size when
something expands, not to the actual sizes before and after. I'll try and

The coefficient of LINEAR expansion is the increase in length, per unit
length, for a rise of one degree temperature. The coefficient of CUBIC
expansion is the increase in volume, per unit volume, for a rise of one

Let's suppose we have an exact one-inch cube of material, at room
temperature, say 20 degrees C. Now heat it to 21 degrees C. It expands just
a little, much less than a thousandth of an inch. For this discussion, let's
suppose it expands exactly one thousandth of its length, or 0.001 of an
inch. We'd say the coefficient of linear expansion for this stuff is 0.001
per degree C. (A real value, for a typical fired clay, would be more like
0.000007 per degree. Not much!)

Of course the cube expands in all directions. The East-West side increases
by 0.001, and the North-South side too, and the Top-to-Bottom side as well.
So the cube swells up. The volume increases, by an amount which tells the
coefficient of CUBIC expansion.

Now, if you're a left-brain thinker (maths, numbers, calculators) you'd
figure it like this:

The new volume is 1.001 x 1.001 x 1.001 = 1.003003

So the coefficient of cubic expansion (the increase in volume) is 0.003003
per degree C. You could be excused for rounding this down to 0.003, which is
just three times the value of the linear coefficient. And this rounded-off
value is what's commonly used.

If you want to get into cube roots instead, you'd have to take the cube root
of 1.003003, to get 1.001, to work this out backwards. But you would NOT
just take the cube root of 0.003003

If you're a right-brain thinker (images, pictures, visualizing shapes) all
this would be easier to explain with a drawing. But think of the cube as it
expands. The top surface rises up by a thousandth of an inch, so adding a
thousandth of a cubic inch to the top. The front face advances towards you
by the same amount, and adds another thousandth of a cubic inch. And one of
the side faces... say the right side... advances and adds a further
thousandth of a cubic inch. Total increase is three thousandths of a cubic
inch. So the coefficient of cubic expansion (the increase in volume) is
0.003 per degree. Of course we've neglected that microscopically small bit
along the edges where the expanding sides aren't really 1 inch wide any
more, but a thousandth of an inch bigger. Neglecting these tiny edges is the
same approximation as rounding down the number value above, from 0.003003
to just 0.003 per degree.

Published values for expansion coefficients have to be looked at closely.
Sometimes the author states the linear value, sometimes the cubic value.
Sometimes it's the expansion per Celsius degree, sometimes per Fahrenheit
degree. Sometimes just gives a figure and doesn't say which one. But David
Hewitt has covered all that so well. If you're not asleep by now, go look at
his excellent web pages.

Roger Graham