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Many studies indicate that the cascade liquefaction process has the lowest energy consumption, and among mixed-refrigerant processes, the two-stage mixed-refrigeration approach is more energy-efficient than the single-stage version. I wonder what the theoretical basis behind this is? I understand that in simulated calculations, the reduction in energy consumption is ultimately reflected in whether the temperature difference across the hot and cold streams at the pinch points is uniform (this can be checked in the HYSYS cryogenic tank performance/plots). Am I misunderstanding this? In a cascade system, different pure refrigerants are used for stage-by-stage cooling; should the resulting curve be step-shaped, similar to the heat exchange curve of propane multi-stage cooling? The curve obtained from mixed-refrigerant cooling should, under ideal conditions, be close to two parallel curves, and its efficiency should be higher than that of a cascade system ； For both two-stage and single-stage mixed refrigeration processes, although there is a difference between single-stage and two-stage cycles, the fitted curves together should form approximately one curve, with that curve being divided into two segments. I’ve checked many papers, and the results are all the same. I’m not sure where I went wrong; I hope experts can help analyze this. Thank you!