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 × 10^{5 }N/mm^{2} and Poisson's ratio is 0.3. Take bulk modulus of water like 2 × 103 N/mm^{2}.
Solution
Diameter of the vessel, d = 750 mm.
Thickness, t = 8 mm.
Net increase in volume, dV = 1000 cm^{3 }= 1 × 106 mm^{3}.
Let enhance in pressure be p N/mm^{2}.
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 × 10^{5}) (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× 750^{3} )/6
= 2.2089 × 10^{8} mm^{3}
Increase in the capacity of the vessel,
δV_{1} = ε_{v} × V
= (2.4609 × 10^{- 4} ) p × 2.2089 × 10^{8} = 54359.6 p
Original volume of water in the vessel = 2.2089 × 10^{8} mm^{3}
Increase in pressure = p.
∴ Decrease in volume of water, δV_{ 2} = (p / K )V
= p/ (2 × 10^{3}) = 2.2089 × 10^{8 }= 110445 p
Net volume change (added volume of water pumped in)
= Increase in capacity of vessel + Decrease in volume of water
δV = δV_{1} + δV_{2}
1 × 10^{6 } = 54359.6 p + 110445 p
∴ p = 6.06 N/mm^{2}.