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Methods of calculating the heat leak through a vapor-cooled support tube of a cryogenic storage vessel were examined. A rigorous method of calculation was developed and its predictions were compared with limiting solutions and where possible with experiment. For the most part, the calculations were performed for an existing high-pressure liquid hydrogen storage vessel constructed of stainless steel, and it is concluded that for such systems the heat leak through a vapor-cooled support tube will probably be insignificant. Calculations for stainless steel and aluminium alloy vessels, containing the cryogens helium, hydrogen, nitrogen or oxygen, indicate that in a vessel constructed of an aluminium alloy the heat leak through the neck will be appreciable. (Author)