Technical Notes on the VPM

 

 

 

Copyright 1999 by Eric Maiken

 

A series of notebooks prepared for the Reverse Diving Profiles Workshop can be found here

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.

Even if you don't have this paper it's ok. The results are in the plots below.

The equations describe how a gaseous bubble responds to the application of hydrostatic pressure. Of course, the bubble shrinks....  But, the quantitative description of the decrease in bubble radius requires consideration of the disolved gas tension surrounding the bubble, as well as the influence of the bubble's skin on both the flow of gas and the bubble's internal pressure. 

The term "varying permeability"  refers to two different pressurization regimes:

1) Permeable A relatively low pressure region (less than 8-10 atmospheres), where gas can freely flow through a skin of surfactants into or out of bubbles during compression and decompression.

2) Impermeable The higher pressure region (greater than 8-10 atmospheres), where gas cannot flow through the skin during compression, but can flow during decompressin.

In laboratory experiments (and presumeably in decompressing divers), the number of bubbles that grow after decompression is related to the depth of the dive (hydrostatic pressure) through a characteristic radius r0, which is the cut-off between growing and shrinking bubbles in the distribution of bubble sizes. 

 

The onset of impermeabllity was set at 9.2 Ata. The pressure units fsw = feet-of-salt water. 3.3 fsw = 1 meter salt water.
     
rVSDepth.gif (10143 bytes) Fig. 1 A plot of the reduction of bubble radii with dive depth (pressure) for six different initial cases.

The family of curves is for bubbles that start out on the surface with a range of initial radii: 0.7< r0 < 1.2 um
PssVSDepth.gif (10899 bytes) Fig. 2 A plot of the minimum supersaturation gradients vs. dive depths for the six different initial bubble radii considered above. In principle, for a given depth and a fixed bubble radius, one point models the allowed supersaturation gradient for every compartment over the entire decompression.

The general result is: the larger the bubble, the smaller the allowed supersaturation. This is why the VPM program becomes more conservative if you change the hard-coded r0 from 1.0 microns to say 1.2 microns.

Iterations following the Yount and Hoffman algorithim will cause a dispersion in the allowed supersaturation. However, for long, deep dives this dispersion is negligeable.

 

You can see the onset of impermeability causes a kink at 9 Ata. The allowed supersaturations become extra conservative for deep dives. This is because the nuclei haven't been crushed as much as one would expect from the permeable form of the model.

Note also that at the onset of impermeability, these curves deviate from the conventional form:

PSATURATION = constant1xP1stSTOP + constant2 .

 

Fig. 3 In equation 19b, an approximation was made, which sets the compartment tension equal to atmospheric pressure for a rapid compression to the impermeable regieme.

However, operationaly, 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. This ingassing information can be used to remove the approximation in a numerical solution that finds the roots of 19b.

The counter-intuitive result is that the faster compartments have larger radius bubbles than the slow compartments . This is physically plausible because faster compartments will have higher tissue tensions, which drives diffusion 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?

You can see this in Fig. 3 by noting when the rainbow of curves crosses the horizontal (radius) axis.These are the roots of eq 19b. Each curve is a separate solution to eq 19b after a 5 minute descent to 9.2 Ata on a trimix. The fast compartments (hot colors) have larger radii than the slow compartments (0.33 microns as compared to 0.3 microns)