Quantitative Measures of Risk.

The potential energy stored when compressing the water is an important useful metric for evaluating risk in the event of a catastrophic failure of the PTV. In Appendix B, the following expression is derived for the potential energy stored in a volume $V$ of compressed water:

  $$E(p) = \frac{\gamma}{2} V p^2~.$$ (2)

The quantities $p$ and $\gamma$ are the pressure and the compressibility of the water, respectively.

Definition of “risk”. Equation (2) does not measure the probability of containment failure -- the storage of more energy does not necessarily imply that containment failure is more likely. In fact, likelihood of containment failure is expected to be in proportion to the pressure. However, (2) does measure the hypothetical destructive capacity associated with containment failure and in this sense is a measure of risk. The direct association of “risk” with “destructive capacity” (as opposed to the probability of containment failure) will be adopted for the remainder of this article unless otherwise noted.

Equation (2) exposes three very important insights.

1. The potential energy stored is proportional to the mean compressibility of the fluid. This explains and quantifies the intuitively obvious notion that highly compressible fluids like gases pose a more serious risk. For pressures up to 10 kpsi, gases are $10^4 \longrightarrow 10$ times more compressible than water and so gases store several orders of magnitude more energy than the same volume of water at a given pressure. For this reason, it is important to bleed all of the air from the vessel prior to pressurization.

2. The stored energy is proportional to the volume of the water in the vessel. If the full volume of the vessel is not required for a particular operation, the risk can be measurably reduced by simply installing the (incompressible) solid aluminum ingots into the vessel.

3. The potential energy stored in the compressed water is proportional to the square of the pressure. For example, doubling the pressure causes the stored energy to increase by a factor of four. This strong dependence of risk on pressure should be explicitly considered when formulating plans for a particular operation.

To appreciate the quantities of energy that are involved, consider the following cartoon scenario. Suppose that the vessel failed at a pressure of 10 kpsi in such a manner that all of the stored potential energy were used to propel the 3650 lbs vessel cap straight upwards into unobstructed space. The volume of the vessel is $V \doteq 4.5 \cdot 10^4~\mathrm{in}^3$. According to (2) the amount of stored energy will be

$$E = \frac{2.9 \cdot 10^{-6}~\mathrm{(psi)}^{-1}}{2} \cdot ... ...12~\mathrm{in}} = 5.4 \cdot 10^5~\mathrm{ft}\cdot\mathrm{lbs}$$

A half-million foot-pounds of energy is enough to propel the 3650 lbs cap approximately $H = \frac{5.4 \cdot 10^5~\mathrm{ft}\cdot\mathrm{lbs}}{3650~\mathrm{lbs}} = 150~\mathrm{ft}$ straight upwards. This cartoon does not suggest what actually would happen as a consequence of containment failure; it merely presents a ready visualization of destructive capacity.

Figure 2 is a graphical expression of equation (2) where each curve represents a different volume of water. The pressure vessel has three solid aluminum ingots that can be used to reduce the volume of compressed water to $\frac{3}{4} V$, $\frac{1}{2} V$, or $\frac{1}{4} V$. The figure demonstrates that the vessel at quarter-volume and 10 kpsi stores the same energy as when at full-volume and 5 kpsi. Although the destructive capacity is the same for each case, the probability of containment failure due to crack propagation at 10 kpsi is roughly quadruple what it is at 5 kpsi.

Figure 2: This figure presents the quantity of potential energy stored in the compressed water as a function of pressure and the volume of water in the vessel. The volume of the vessel is $V$ but solid aluminum ingots can be used to reduce this volume to $\frac{3}{4} V$, $\frac{1}{2} V$, or $\frac{1}{4} V$. The right ordinate axis expresses the potential energy in conventional terms while the left ordinate axis expresses the energy in terms of the hypothetical vertical distance that the 3650 lbs cap could be propelled. The quiet protocol applies to pressures less than 3500 psi, the notify protocol applies in the yellow-shaded region, and the evacuate protocol applies elsewhere.

Figure 2 is especially useful for evaluating the marginal risk associated with increasing volume and pressure either separately or together. For example, the risk associated with operating the half-volume vessel at 6 kpsi is only 18% of what it is at full volume and 10 kpsi. That is, 82% of the total destructive capacity of the PTV is associated with the upper 40% of the pressure range and upper 50% of the volume range.