Hydraulic Expansion Pressure Rise
API Standard 521, 6th Edition, §220.127.116.11.1 provides an equation (shown below) for the calculation of the pressure rise due to hydraulic expansion of a non-boiling liquid1. The calculation is comprised of four basic components: the net volumetric expansion of the container and the liquid, the reduction of volume due to leakage, the compressibility of the liquid itself, and the stress-strain response of the container. When investigating the basis for this equation, we found some variations in different sources regarding the stress-strain response of the container, which is worthy of clarification.
The CCPS Guidelines for Pressure Relief and Effluent Handling2 provides a more convenient expression for the pressure rise calculation for the purposes of discussing the stress-strain response, noting that the ‘leakage’ term is absent. In comparing the equations, one finds differences in the container strain correlation, labeled ‘c’ in our rearrangement above. Various sources, including references  and , have different equations for ‘c’ (as a function of Poisson’s ratio µ), which prompts the need for clarification:
- API Standard 521, 6th Edition1: c = ½ (2.5 – 2 µ)
- CCPS Guidelines, 1st Edition2: c = 1 – ½ µ
Our investigations indicate that the value for the container strain correlation is dependent on the geometry of the container (i.e. cylindrical or spherical shape as well as thin-walled or thick-walled), supporting (i.e. restraining) of that container, and material of construction (e.g. isotropic vs. anisotropic behavior). This correlation is a function of the ratio of the lateral to axial strain (Poisson’s ratio, µ, usually having a numerical value of approximately 0.3) and the ratio of the container diameter to the container wall thickness. With respect to the effect on the pressure rise calculated, a conservative estimate for this correlation would be 0. Discussions at the API Subcommittee for Pressure Relieving Systems many years ago indicated that the correlation published in API Standard 521 was specifically for a vertical cylindrical vessel anchored to a foundation. Other sources provide correlations for other situations:
- Copenhaver, Coppari, and Rochelle3: c = ¾ (1 – µ) for thin-walled piping
- Streeter and Wylie4 suggest the following for thin-walled containers having a diameter to thickness ratio greater than approximately 40:
- c = (1.25 -µ) for containers free to stress and strain both laterally and longitudinally yet anchored at only one point
- c = (1 – µ²) for containers rigidly anchored to prevent axial strain
- c = 1 for containers having fully functioning expansion joints
Since the evaluation of the pressure rise due to hydraulic expansion is not usually performed — most simply accept the common position that a near infinite pressure rise can occur — the distinction between these correlations would not appear to be significant; however, when this calculation is employed, it is usually done for special cases. It is for these special cases that the distinctions should at least be understood.
 American Petroleum Institute. “API Standard 521: Pressure-relieving and Depressuring Systems”. 6th Edition, 2014 Jan.
 AIChE Center for Chemical Process Safety. “CCPS Guidelines for Pressure Relief and Effluent Handling Systems”. 1st Edition,
 Streeter and Wylie, Hydraulic Transients, 1967.
 Copenhaver, Coppari, and Rochelle, “Forestall Pipe Bursts”, Chemical Engineering, January 2001, pp. 84-89.