Amount/flow rate of N2 needed to cool substrate

I'm trying to cool a substrate inside a vacuum chamber with liquid nitrogen. The substrate holder is a block of copper suspended from the top of the chamber via nylon screws (for thermal insulation). A copper tube runs from the LN2 tank through the flange, through one of the substrate holder's through-hole, bend back through the other substrate holder through-hole, and exits the chamber. Upon exit, the LN2 is simply dumped into the atmosphere.

The substrate holder starts at 288.2K, LN2 ~77K. The LN2 tank's line pressure is around 5 psi, and ends in standard pressure upon exit.

I'm trying to calculate whether with my current setup I just described, I can actually cool the substrate (ITO coated with aluminum, with a foil of indium between it and the substrate holder surface) down to 77-90K.

My problems right now are setting up the diff eqs describing the heat exchange inside the substrate holder. The flow rate of N2 inside the pipes depend on whether it would be heated to gas...I'm not exactly sure how to incorporate this into the diff eqs.

I would really appreciate it if someone can walk me through how to approach this or point me to a sample calculation.

Also, what would be the best way to achieve this kind of cooling? We've used a reservoir approach (having a vat of LN2 on top of the substrate holder) before with success, but it takes up a lot of space. If my current setup turns out not sufficient, can I just keep increasing the initial line pressure (to >5psi) to achieve the desired cooling?

Thanks, Al

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This started as a comment that grew beyond all reason.

You can assume the LN2 will evaporate until the temperature of your substrate gets reasonably close to 77 K [1]. Typically you know--empirically--that such a system is cold when you start to get liquid drops coming out the exhaust. Then you turn down the flow rate and monitor the temperature with your in situ thermometer.

You do have in situ monitoring, right? If not you're going to have to do this at least partially empirically. What is your tolerance for letting the widget warm up?

In any case a vacuum system and nylon stand-offs is a good start for having it insulated, the copper pipe needs to be kept cold or it will conduct a lot of heat, but you'll be flowing LN2 in it, no?

Sounds like it should work.

Doing a full-blown calculation from first principles is a lot of work, and it might be better to find an existing simulation framework [2]. The scientific computing beta site might help there.

[1] This suggests a guess at the minimum LN2 needed for initial cooling in terms of the latent heat of vaporization for LN2: $m_{s} c_{s} \Delta T \approx m_{N} L_N$ (where $s$ is the substrate and $N$ is LN2). Depending on losses you might need twice that.

[2] We did that for beam related heating of solid targets at JLAB because melting one meant a week or more of lost beam to replace it; but there was a staff engineer responsible for the program.

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Thanks for the quick replies! Indeed I do have in-situ thermometer. The lowest I've seen from the setup so far is 132K. I've also seen liquid drop out of the exhaust, so it might suggest some contact problem of the in-situ thermometer. – Asy May 25 '12 at 0:11
Yeah. That or insufficient thermal contact between the LN2 pipe and and the subject. The one target we did melt at JLAB was on account of mediocre contact with the cooling system. – dmckee May 25 '12 at 0:18