Determine the increase in pressure in vessel:

A thin spherical vessel of diameter 750 mm & wall thickness 8 mm is filled through water at atmospheric pressure. Determine the increase in pressure if an additional volume of 1000 cc of water is pumped into the vessel. Young's Modulus of the material of the vessel is 2 × 10⁵ N/mm² and Poisson's ratio is 0.3. Take bulk modulus of water like 2 × 103 N/mm².

Solution

Diameter of the vessel, d = 750 mm.

 Thickness, t = 8 mm.

Net increase in volume, dV = 1000 cm³ = 1 × 106 mm³.

Let enhance in pressure be p N/mm².

Permitting for the compressibility of water, the added volume of water pumped in is the net change in volume because of enhance in the capacity of the vessel and reduces in the volume of water.

Hoop stress, σ  = pd/4t = p × 750/4 × 8 = 23.4375 p

Hoop strain, ε _h  = (σ_h / E )(1 - v) = (23.4375 p/2 × 10⁵) (1 - 0.3) = 8.203 × 10^{- 5} p

Volumetric strain, ε_v = 3 × ε_h

= 3 × (8.203 × 10^{- 5} ) p = 2.4609 × 10^{- 4}  p

Original volume, V = π d ³ / 6

=( π× 750³ )/6

= 2.2089 × 10⁸  mm³

Increase in the capacity of the vessel,

δV₁ = ε_v × V

= (2.4609 × 10^{- 4} ) p × 2.2089 × 10⁸  = 54359.6 p

Original volume of water in the vessel = 2.2089 × 10⁸  mm³

Increase in pressure = p.

∴ Decrease in volume of water, δV ₂   = (p / K  )V

= p/ (2 × 10³)  = 2.2089 × 10⁸ = 110445 p

Net volume change (added volume of water pumped in)

= Increase in capacity of vessel + Decrease in volume of water

δV = δV₁ + δV₂

1 × 10⁶  = 54359.6 p + 110445 p

∴          p = 6.06 N/mm².