Calculating the soak time for a solid cylinder immersed in liquid nitrogen. 3

[KathrynM]

Hello,

In order to temporarily shrink an axle to accommodate an interference fit with another component, I am trying to calculate the soak time reguired for the axle to reach a temperature equilibrium with the liquid nitrogen in which it will be immersed.

The axle is solid and cylindrical in shape and its length is short in relation to its diameter. As the temperature will vary as a function of time and both axial and radial position (I think), I am guessing that some 2-dimensional transient analsyis is required. The heat equation corresponding to these conditions doesn't exactly look straightforward to solve. I have a book on the fundamentals of heat and mass transfer, which doesn't consider the solution of the 2D transient heat equation but does analyse multidimensional transient conduction for a range of geometries. However, a heat transfer coefficient (h) is stipulated for all of these cases. I don't have this, and I gather that it can be worked out from the heat flux. However, isn't the heat flux worked out from the expression for temperature ...which I don't have? I might be missing something obvious as it has been the best part of a decade since I did any thermodynamics.

If anyone could help me out with this it would be much appreciated as I'm going round in circles.

Thanks.

Kathryn.

 

[boyze]

In some textbooks of HT fundamentals(Incropera and Dewitt for example) they discuss special cases of radial transient conduction of infinitely long solid cylinders that are "suddenly" immersed into a hot or cold fluid. The trick is to assume that the surface is held at constant T, i.e., h_c is infinite. Curves are provided that can give T(t). This approach would provide an upper bound on time given the shaft length in your case. The textbook case assumes no axial conduction.

If you need to be more precise then the axial conduction would need to be included. An FEA or FD model could be done. Or use the upper bound value and do a test. Depending on the amount of assembly clearance needed (factoring in tolerance scenarios) you may not need a full chill anyhow.



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James A. Pike

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[Compositepro]

Why do you need to calculate this? The heat flux into the cylinder can be pretty easily visualized by the observing the rate of boiling of the LN2. When boiling stops the temperature is at equilibrium.

 

[MintJulep]

The heat transfer coefficient of the boiling interface between metal and Nitrogen will be interesting.

Experimentally you could perhaps make a reasonable swag knowing the heat of vaporization and the mass reduction of liquid from the bath in a known time.

 

[racookpe1978]

Are you really, really sure that you need to chill that cylinder down to liquid N2 temperatures just for an interference fit?

Solid CO2 as little pellets - not big blocks! - is usually cold enough to get the part down to the right size, much safer to handle, and is much more easily (cheaper!) obtained.

 

[ione]

As noted by racookpe1978 dry ice (solid CO2) is listed among thermal vectors to induce cold shrinkage. If the temperature of -109 F (approx -78 °C) is not enough in order to achieve the required contraction the use of LN at -321 F (-196 °C) is the way to go in order to get further shrinkage.
The Machinery’s Handbook reports the following values of shrinkage per inch of diameter for a temperature reduction from +75 F to – 321 F (deltaT = 220 °C):
1) for steel from about 0.002 to 0.003 inch for steel;
2) for aluminum alloys 0.0042 inch;
3) for magnesium alloys 0.0046 inch;
4) for copper alloys 0.0033 inch;
5) fro monel metal 0.0023 inch;
6) for cast iron (not alloyed).

 

[ione]

If you have to deal with a short cylinder you can consider a combination (product) of the solution relevant to two 1-d scenarios that is plane wall and infinite cylinder, whose intersection gives your 2-d configuration.
In order to compute the Biot number, which allows you to get a couple of coefficients needed for the two scenarios mentioned above, you can assume a heat transfer coefficient for LN (natural convection on a cylinder) on the order of 300 W/(m^2*K). If you want you can play with numbers reported in the attached file.

 

[chicopee]

Wouldn't it be easier to heat up the other component to which the axle is to fit?