Source term saturated liquid flow rate gap

It is possible that most source term models overestimate saturated toxic chemical liquid source term release rates by 2 – 3 times, depending on the circumstances. 

Table 1:  Sozzi and Fauske experimental saturated water flow rate data compared against various tube lengths

The left hand portion of Table 1 shows the experimental flow rate of a container with a tube length of 73 cm and compares that to a theoretical release with a tube length of 0 cm using the Bernoulli equation (Eqn. 1), which takes gravity and pressure differences into account.  The right hand portion of Table 1 shows experimentally determined flow rates based on various tube lengths.  In both cases the saturated liquid flow rate is dramatically affected by the length of the tube leading from the container to the ambient atmosphere. 

                                                                                                            (Eqn. 1)

            Equation 1 is appropriate if the liquid doesn't flash in the container or the tube, creating a choked flow condition whereupon increasing the pressure differential between the container and the ambient surroundings doesn't increase the saturated liquid flow rate.  If a saturated liquid is released through a tube longer than 10 cm, it is generally understood that the fluid will partially flash and equilibrate into a stable two-phased (or fully choked) condition, resulting in a mass flow rate that can be 30 - 45% of the value given in Equation 1. 

            Real world containers have a wall thickness or effective tube length between 2.5 – 8 cm and don't fit neatly into the straight-edged orifice category described by the Bernoulli equation or into the long tube lengths that develop into fully developed two-phased choked flow.  This presentation will highlight a discrepancy between real world containers and source term models, explaining the effect that a container wall thickness greater than 0 cm and less than 10 cm has on the source term flow rate of a saturated liquid, and by extension the effect it has on the chemical dispersion and human injuries.

            Experimental evidence for saturated liquids released into the ambient environment suggests that the Bernoulli equation may only be applicable with a truly straight-edged orifice and that even short tubes can cause liquids to flash, creating a partially choked flow condition.  The problem is that every source term model we have encountered uses the Bernoulli equation to describe saturated liquid flow and ignores the fact that real world containers have a wall thickness, which will reduce the flow rate of the fluid.  Figure 1, in the associated table, gives the experimentally determined flow rates as a fraction of Bernoulli flow for various tube lengths.

Figure 1:  Mass flow rate results for saturated water using various tube lengths

            Saturated chlorine is often transported by railcar, and a railcar has a 2.5 cm thick steel plate along with 10 cm of ceramic and fiberglass insulation, for a total wall thickness of 12.5 cm.  Any breach in a real railcar will result in two-phased flow with a dramatically reduced mass flow rate, but current source term models assume a straight–edged container and use Equation 1 for railcar release modeling. 

            Taking wall thickness into account could improve the accuracy of source term models by 2 – 3 times by providing a more accurate source term flow rate.  There are times when it is sufficient to bound a source term problem using the “Bernoulli equation” for liquid flow (maximum flow rate) and the “Omega method” for fully developed two-phased flow (minimum flow rate).  There are also times when it is important to obtain a more accurate answer to a source term problem.  The arguments in favor of accounting for wall thickness in source term models are:

1.      Accounting for wall thickness when modeling a saturated liquid release will give a more accurate source term release rate.

2.      Monte Carlo modeling could require half the computer time if the source term flow rate were one variable rather than bounding the problem using the Bernoulli equation and the Omega method. On the other hand, even when bounding the problem is desired, an additional run with wall thickness taken into account could provide more realistic expected source term.

3.      A semi-empirical equation accounting for saturated liquid release rates through a 0 – 10 cm tube already exists and has been proven against existing water, Freon-11, and Freon-12 data.

4.      Scientists and engineers could study wall thickness affects as it pertains to public safety, plant design, and vessel design.

5.      During an actual release, if the chemical being released was known, then using the standard wall thickness equation it would be easy to calculate the thickness of the container wall in a realistic fashion and take that variable into account when modeling the source term flow rate.