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In hadron colliders such as the LHC, the energy deposited in the superconductors by the particles lost from the beams or coming from the collision debris may provoke quenches detrimental to the accelerator operation. In previous papers, a Network Model has been used to study the thermodynamic behavior of magnet coils and to calculate the quench levels in the LHC magnets for expected beam loss profiles. This model was subsequently used for thermal analysis and design optimization of quadrupole magnets, which LARP (US LHC Accelerator Research Program) is developing for possible use in the LHC luminosity upgrade. For these new magnets, the heat transport efficiency from the coil to the helium bath needs to be determined and optimized. In this paper the study of helium cooling channels and the heat evacuation scheme are presented and discussed.