PSI - Issue 65

Emelyanov I.G. et al. / Procedia Structural Integrity 65 (2024) 83–91 Emelyanov I.G., Puzyrev P.I. / Structural Integrity Procedia 00 (2024) 000–000

88 6

hydrogen should leave a vessel with a wall thickness of 1 cm. However, this does not happen: hydrogen, even in thin-walled cylinders, lasts for years.” Therefore, the correct formulation of the initial and boundary conditions and the physical constants accepted in the calculations will determine the adequacy of the resulting non-stationary field of hydrogen concentration distribution in the target body. It is advisable to verify the solution results for applied problems by comparing the solution results using various numerical methods. Fick's second law, Vorobyov (2003) and the heat transfer equation have much in common, since both describe the process of transfer of a substance through a material. Fick's second law describes the diffusion of a substance through a material. And the heat transfer equation describes the transfer of heat through a material. Both equations use the concept of a concentration or temperature gradient to describe the direction of transfer of matter or heat in a material. For the one-dimensional case, the heat conduction and diffusion equations have the form:     2 2 , , T x t T x t a t x      (5)     2 2 , , c x t c x t D t x      (6) where c - hydrogen concentration; x – coordinate; D is the diffusion coefficient, which characterizes the efficiency of the diffusion movement of the substance and has a dimension of m 2 /s. The volumetric concentration of hydrogen in metals c is measured by its content in ppm parts per million (1.12 cm 3 /100g). At room temperature and atmospheric pressure, hydrogenation from a gaseous medium practically does not occur. At atmospheric pressure and T = 500 ºС, the solubility of hydrogen in iron is c = 0.6 cm 3 /100 g, which corresponds to 5.3·10 -5 % of weight. With increasing temperature, solubility increases greatly. At high pressures and temperatures, the solubility of hydrogen in some steels can reach 300-400 ppm. The solubility of hydrogen in titanium can reach 0.002% by weight, which approximately corresponds to 40,000 ppm in room temperature, Cherdantsev, Chernov and Tyurin (2008). In the heat transfer equation, a temperature gradient in a material causes heat to be transferred in the direction of increasing temperature. In the Fick equation (6), the concentration gradient causes the substance to diffuse in the direction of decreasing concentration. It is known that the diffusion process strongly depends on body temperature. Einstein's relation relates the mobility of a molecule with the diffusion coefficient and temperature, Vorobyov (2003): p B D k T   (7) where k B is Boltzmann's constant. Since the diffusion process is much slower (by several orders of magnitude) than the temperature propagation process, the diffusion process is usually divided into time intervals, within which the value of D takes a constant value. Therefore, based on equations (5) and (6), we can imagine: D D k a  (8)

where k D is the reduction coefficient of the diffusion process at T = const . Taking into account relations (1), (2) and (8), the diffusion equation can be written in the form: 1 c c c D rD z z r r r t                        

(9)