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?