Estimating Fluid Temperature for Contraction of a Metal Cylinder

In the contraction of a hot steel cylinder immersed in a cooler medium, the coefficient of thermal expansion of steel decreases with temperature. This Demonstration shows how to calculate the temperature of fluid needed to get a predetermined contraction.

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Immersing metal parts in a cool medium is used consistently in shrink-fit assemblies. To calculate the contraction in these metal parts, one needs to know the thermal expansion coefficient of the metal, the original dimensions of the part, and the temperature change. For example, if one immerses a solid cylinder at room temperature in a cool medium such as dry-ice/alcohol or liquid nitrogen, then the reduction in the outer diameter of the cylinder may be calculated using
where
the outer diameter of the cylinder,
is the coefficient of thermal expansion at room temperature,
is the temperature of cool medium, and
is the room temperature.
However, the coefficient of thermal expansion of metals such as steel varies considerably with temperature, as shown in the table:

The contraction of the diameter of the cylinder for the case where thermal expansion coefficient varies as a function of temperature is given instead by
One way to proceed is to find the contraction in the diameter is to curve-fit the coefficient of thermal expansion versus temperature data by using polynomial regression. In this case, we can fit a second-order polynomial
Then
,
If the amount of contraction, , initial temperature, , and the diameter, , are known, we need to solve a cubic equation to find the temperature of the fluid, , that is needed to give the required contraction.
References
[1] A. Kaw and A. Yalcin, "Problem-Centered Approach in a Course in Numerical Methods," ASCE Journal of Professional Issues and Engineering Education, 134(4), 2008 pp. 359–364.
[2] A. A. Kaw, D. Nguyen, and E. E. Kalu, Numerical Methods with Applications, 2010. http://autarkaw.com/books/numericalmethods/index.html.
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