AbstractHeat transfer from a cylindrical copper rod to He I or to saturated He II confined in a narrow duct was investigated to obtain the conditions for a cryostatic stabilization of superconducting magnets.The heat transfer characteristics under a quasistatic sweep of the input power are enhanced by flow induced by convection and by the chimney effect in vertical sections of the channels. In addition to these self-induced flows in saturated He II, a cyclic vaporization above the critical heat flux of superfluidity can impede the onset of a transition to the film boiling state. However, such an enlarged peak heat flux to the film boiling state Qp does not give a reliable upper limit for superconducting composites confined in narrow channels where a self-induced flow cannot be quickly activated. Here a transient heat input with a threshold smaller than that of the steady state can trigger a transition. For an effective stabilization, a critical channel configuration must be maintained by retaining sufficient space for the coolant in the superconducting windings.