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 $168\mathrm{.}3mm(6\mathrm{.}625$ in$.),$ a wall thickness of $3\mathrm{.}4mm(0\mathrm{.}134$ in$.),$

and a solid cross$-$sectional area of $17\mathrm{.}63c{m}^2(2\mathrm{.}733i{n}^2).$ The inner line $(2)$ is a

$100-$mm nominal $(4-$in$.$ nominal$)$ SCH $10$ pipe with an outside diameter of $114\mathrm{.}3$

$),$ a wall thickness of $.),$ and a solid cross$-$sectional

area of $10\mathrm{.}65c{m}^2(1\mathrm{.}651i{n}^2).$ The length of the outer line is $12\mathrm{.}00m(39\mathrm{.}37ft),$ and

the length of the inner line is $12\mathrm{.}20m(40\mathrm{.}03ft).$ The system is stress$-$free at a

temperature of $20C(68F).$ Under operating conditions, the temperature of the

inner line is $-180C(-292F),$ and the temperature of the outer line is

F$).$ Both lines are constructed of $304$ stainless steel, with Young's modulus, $198$

MPa$(28\mathrm{.}7106)$ and thermal expansion coefficient,

$\{$:${10}^{-6}\frac{1}{F}).$ Because of other loadings, it is desired to limit the thermal stresses for

the system to $16$MPa$(2320)$ by placing an expansion bellows in the outer line.

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.