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This paper presents a study of the temperature distribution on a universal cryogenic surface when it is cooled to nitrogen temperatures. Using computer simulation methods, in particular the finite element method, it was possible to determine the temperature-time dependences that occur during the cooling of a surface using the heat exchange tubes of different diameters located inside the surface at equal distances from each other. By varying the diameter of the heat exchange tubes, its effect on the cooling intensity was revealed when examining the volumetric temperature distribution: a larger diameter of the heat exchange tube yields a more uniform cooling of the surface. Besides, a larger orifice diameter leads to reaching thermal equilibrium quicker over the entire surface examined. By using a 7 mm orifice diameter, thermal equilibrium was reached 70 seconds after the start of cooling while 5 mm and 3 mm orifice diameters reached temperature equilibrium after 90 and 100 seconds respectively. The results obtained will make it possible to produce cooling of higher quality when designing universal cryogenic surfaces by using heat exchange tubes with certain geometric parameters. [ABSTRACT FROM AUTHOR]