Problem description: A vacuum-jacketed transfer line for liquid oxygen is shown in the figure below. The outer line (1) is a 150-mm nominal (6-in. nominal) SCH 10 pipe with an outside diameter of 168.3 mm (6.625 in.), a wall thickness of 3.4 mm (0.134 in.), and a solid cross-sectional area of 17.63 cm (2.733 in). The inner line (2) is a 100-mm nominal (4-in. nominal) SCH 10 pipe with an outside diameter of 114.3 mm (4.500 in.), a wall thickness of 3.1 mm (0.120 in.), and a solid cross-sectional area of 10.65 cm (1.651 in). The length of the outer line is 12.00 m (39.37 ft), and the length of the inner line is 12.20 m (40.03 ft). The system is stress-free at a temperature of 20 C (68 F). Under operating conditions, the temperature of the inner line is -180 C (-292 F), and the temperature of the outer line is +30 C (86 F). Both lines are constructed of 304 stainless steel, with Young's modulus, 198 MPa (28.7 x 106 psi) and thermal expansion coefficient, 14.8 10-6 1/C (8.22 10 1/F). Because of other loadings, it is desired to limit the thermal stresses for the system to 16 MPa (2320 psi) by placing an expansion bellows in the outer line. 12.0 m 12.2 m Expansion Bellows m fin Requirements: 1- Derive the necessary equations governing the problem and determine the required spring constant for the bellows. 2- Check the required stiffness with those which are commercially available from the manufacturers' catalogue.