Modelling CO$_{2}$ cooling of the ATLAS ITk Pixel Detector

The Large Hadron Collider (LHC) physics program has been extended to the period 2026-2037 to deliver an order of magnitude more of proton-proton collisions compared to end 2023. To sustain the harsh conditions imposed to detectors in this period (radiation, high occupancy), the current ATLAS Inner Detector will be replaced by a new one using the most recent silicon sensor technologies. One of them is the generalization of the cooling with two-phase CO$_{2}$ flowing in titanium tubes located close to the silicon sensor and associated electronics. The choice of CO$_{2}$ relies on its most favourable thermo-physical properties in boiling state. The radiation damage during the HL-LHC program will be quite harsh and the engineering of the local supports under such extreme conditions requires a deep understanding of the two- phase behaviour of the CO$_{2}$ flowing inside the titanium pipes, and a precise modelling of the heat transfer through the mechanical structure. However, the data available on CO$_{2}$ boiling in channels of small hydraulic diameter (say, below 3 mm) is limited and often affected by too large uncertainties. This enforces the detector designers to include large safety factors and long iterative phases of experimental measurements, which could be sensibly reduced by the availability of reliable models. This thesis starts with a quick description of the ATLAS Upgrade Phase II project, a first outlook on the fundamentals of two-phase cooling technology, widely used to cool particle detectors in high-energy physics (HEP) experiments and discussion about the thermal management requirements of the future ATLAS Pixel Detector. Several custom tools were developed in python language to guide the design and optimisation of the CO$_{2}$ cooling system: simulation of heat transfer coefficient (HTC) and frictional pressure drop along evaporators, flow distribution studies in manifolds and calculation of thermal requirements. In a second phase, the document presents the thermal test setup to measure thermal performances of thermal prototypes for the new detector (ITk) as well as the associated simulation based on Finite Element analysis and Heat Transfer Coefficient modelling. A novel fit method was implemented to extract from the measurements the different parameters of the ITk elements (in situ materials thermal conductivity, manufacturing variability, HTC. . . ). The results are compared to the current CO$_{2}$ model predictions and the discrepancies discussed. A new set of parameters for the CO$_{2}$ model was defined to improve precision by factor two on the effective HTC values for the ITk working conditions and integrated in the python modelling tool. Finally, the impact of this work on the design of the Pixel detector is discussed.