A semi-empirical model for preheater design to trigger CO$_2$ boiling for detector cooling

The fluid properties of CO$_2$ make it an ideal medium for the cooling of tracking detectors in experiments at particle accelerators. Detectors such as the Compact Muon Solenoid Outer Tracker at CERN will be cooled to a nominal temperature of -35$^{\circ}$C with CO$_2$ cooling to ensure the longevity of the silicon sensors. In theory, two-phase CO$_2$ cooling results in a very low temperature change along the detector tube, dependent only on pressure drop. Experimentally, however, superheating - the existence of a fluid in the liquid form above its boiling temperature - has been observed to occur frequently. This results in higher fluid temperatures and a poor heat transfer coefficient over the first section of the detector tube, disrupting the cooling performance of the detector and possibly leading to deterioration of the silicon sensors. In order to prevent superheating, a preheater is proposed to trigger nucleate boiling in the Compact Muon Solenoid Outer Tracker detector cooling tube just upstream of the sensors. A theoretical - semi-empirical - model for the preheater design is presented, starting from experimental data points. With this model, the triggering of nucleation can be characterised for tubes made of the same material as that tested and with the same surface cavity size. The model validation is promising, closely matching the trends from experimental results, and giving preheater specific powers significantly lower than those derived from spinodal theory.