Locked-in thermal pressure

When a vessel or a section of pipe is isolated while filled with a fluid and subject to a temperature increase, there is a corresponding pressure increase.

Symbols used:

$α_{f}$	is the volumetric coefficient of thermal expansion for the fluid (°C^-1);
$α_{s}$	is the volumetric coefficient of thermal expansion of the shell (°C^-1);
$Δ T$	is the change in temperature (°C);
$β$	is the compressibility factor of the fluid (MPa^-1);
$D$	is the inside diameter (mm);
$t$	is the wall thickness (mm);
$E$	is Young's modulus for the shell material (MPa);
$Δ P$	is the change in fluid pressure (MPa);

To calculate this we start with the change in volume of the fluid:

Δ V_{f l u i d} = α_{f} Δ T - β Δ P

The change in internal volume of the shell is expressed as:

Δ V_{s h e l l} = α_{s} Δ T + Δ P \frac{D}{t E}

Equating the volume changes and solving for the change in pressure we have:

Δ P = \frac{(α_{f} - α_{s}) Δ T}{β + \frac{D}{t E}}