The high enthalpy shock tunnel can simulate the free-flow speed above 3km/s. The characteristic of the flow is that the kinetic energy of the high enthalpy stagnation gas is high enough to effectuate high-temperature effects such as dissociation even ionization of fluid molecules. The high enthalpy nozzle converts the high enthalpy stagnation gas into hypervelocity free flow. The flow of the high enthalpy nozzle consists of three distinct flow regions: an equilibrium region upstream of the throat, a non-equilibrium region near the throat, and a frozen region downstream of the throat. The study focuses on the conical nozzle, testing thermochemical non-equilibrium expansion effects under the different expansion angle of the expansion section, the curvature radius of the throat, the throat radius, and the convergence angle of the convergent section. A multi-block solver for axisymmetric compressible Navier-Stokes equations is applied to simulate the thermochemical non-equilibrium flow in several high enthalpy conical nozzles. The significant conclusions of this study contain tripartite. Firstly, the thermochemical non-equilibrium effects are sensitive to the maximum expansion angle and throat radius, but not to the radius of throat curvature and the contraction angle. Secondly, as the maximum expansion angle decreases and the throat radius increases, the flow approaches equilibrium. Finally, the maximum expansion angle and the throat radius not only affect the position of the freezing point but also impacts the flow field parameters, such as temperature, Mach number, and species mass concentration.