VPMechanics 4

Copyright 1999-2001 Eric Maiken

VPM Supersaturation Gradients

This notebook takes a look at the allowable supersaturation pressures (or gradients) generated by the VPM throughout the permeable and impermeable regimes.

Variation of Equilibrium Bubble Radii and Minimum Supersaturation Gradients With Dive Depth

The plots below show solutions to the set of equations (19a-22c) detailed in the article: Skins off varying permeability: A stabilization mechanism for gas cavitation nuclei, D.E. Yount, J. Accoust. Soc. Am. 65, 1429, 1979. For the calculations, the onset of impermeability was set at 9.2 Ata.
Here's a plot of the reduction of bubble radii with dive depth (pressure) for six different cases. The family of curves is for bubbles that start out on the surface with a range of initial radii: 0.7<= <= 1.2 um

Following is a family of the minimum supersaturation gradients vs. dive depths corresponding to the different initial bubble radii considered above. The onset of impermeability produces a kink at 9 Ata, where the trend of allowed supersaturations becomes more conservative for deep dives.

Note that these curves are not the same as conventional plots of the maximum gradients or tensions (M values) allowed at each dive depth.

For a given depth and a fixed bubble radius, one point predicts the minimum allowed supersaturation gradient for every compartment over the entire decompression. For example, if = 1.2 μm is chosen as a parameter, then on a dive to 300 fsw,  the allowed supersaturation would be approximately: Pss =  29.8 fsw. The larger the equilibrium bubble radius, the smaller the allowed supersaturation. This is why the BASIC VPM program becomes more conservative if you change the hard-coded from 1.0 micron to, say, 1.2 microns.
This minimum value is used as a starting point in the calculation of a set of larger supersaturations allowed for each compartment.

The Maximum Supersaturation Gradients

See VPM9 for an update to this section

In the permeable regime, the Y&H algorithm's iterated calculations of an ascent profile causes a dispersion in the allowed supersaturation gradients across the compartments. The fixed Pss, applied to all compartments on the first iteration, relaxes to PssNew on subsequent iterations, with shorter decompressions producing larger separation of the PssNew across compartments.
The following figure illustrates the case for the ZHL16 set of half-times on the final iteration with = 1 micron. This figure applies to no-stop diving, and illustrates the maximum dispersion of PssNew across compartments. On the other hand, there is no dispersion for saturation dives, and all compartments are governed by the fixed supersaturation Pss, which follows the purple curve calculated for the slowest compartment. Note that with the unit conversion 33 fsw = 1 atm, this curve is identical to the = 1micron plot of the previous figure.
Also, compare the figure's values to the PssNew calculated in VPMechanics1 and VPMechanics3 for 100 ft (P = 3.03 atm), and 75 foot (P = 2.27 atm) dives.

The Inverse Gradient Bubble Model?

A somewhat tentative --if unconventional-- result follows from the following analysis. David Yount's cautionary remark can be found  here
In Yount's equation 19b, an approximation was made, which sets the compartment tension equal to atmospheric pressure for a rapid compression to the impermeable regime. However, operationally, it can easily take 3 -10 minutes for a diver to reach 9 atmospheres. During this time, the compartments ongas at exponential rates, which are tracked by decompression programs. Quantitative modeling of this ingassing can be used to remove the approximation in a numerical solution that finds the roots of 19b.
As an example, a dive is made to 400 ft using a 10%a 60%He trimix, with = 1 micron. The decent to 270 ft (9.2 Ata), where the onset of impermeability is assumed to occur, was set at 5 min.
The set of radii r that determine the allowed supersaturation gradients are found from the points where the rainbow of curves cross the horizontal (radius) axis. These are the roots of eq 19b, with each curve a separate solution to eq. 19b for one of the ZHL16 compartments. The fast compartments (hot colors) have larger radii than the slow compartments. Note that for an instantaneous descent, all of the curves would lie on top of the purple (635 min) curve. This curve's zero crossing value of r = 0.295 microns is identical to that found in the first figure of this notebook.

A tabulation of the radii and allowed supersaturation gradients vs. ZHL16 compartment follows.
The counter-intuitive result is that the faster compartments have larger equilibrium radius bubbles than the slow compartments. This is physically plausible because faster compartments would have higher tissue tensions, which drives diffusion of gas into the bubbles.
So? Well, larger bubbles mean smaller allowed gradients --and when have you ever heard of a fast  compartment having a lower G or M than a slow compartment?