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 (28.7 106 psi) and thermal expansion coefficient, 14.8 106 1/C (8.22 106 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. Expansion Bellows 12.0 m 12.2 m 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.