The calculation of ice thickness that can be developed on a section of cryogenic piping will be difficult to calculate, especially when the outside ambient temperature is warmer than 0°C. Consider this as a cylindrical heat gain problem with a composite wall where the following resistances are involved:

• inside heat transfer resistance of cryogenic fluid (probably a high coefficient - minimal resistance)
• conduction through metal wall of piping
• conduction through insulation (is there any?)
• conduction through the ice
• outside resistance—ambient condition dependent—wind (forced convection), no wind (natural convection) or rain (very high film coefficient)

The evaluation of interface temperatures can be calculated directly in this cylindrical conduction problem if you know all the thermal conductivities. Unfortunately, the conductivity of the ice will be dependent upon the volume fraction of air entrapped. If you assume the thermal conductivity of pure ice, then you can solve the problem fairly simply. See "Incropera & DeWitt -- Fundamentals of Heat & Mass Transfer" for the solution, or if you have TK Solver Heat Transfer models, it is one of the models included.

The prevention of ice formation is accomplished by properly sizing the insulation required to maintain greater than 0°C surface temperature as well as providing insulation between the pipe and the pipe supports.