VPMechanics9

Copyright 1999-2001, Eric Maiken

The Dynamic Critical Volume Method for a Single, No-Stop Dive

This notebook revisits the appendix of Yount and Hoffman's 1986 paper.

Compared with earlier notebooks in this series, a major revision of the supersaturation gradients results from the use of proper, non-zero ascent times. For non-stop decompression dives, this effect is contained in the original Yount and Hoffman algorithm.

The aim here is to continue to look into the dynamics of the YH* formulation by exploring the consequences of using a set of parallel compartments to model gas tension surrounding VPM nuclei. In particular, I'm looking for responses from the VPM and dynamic critical volume equations that carry over into making practical recommendations for ascent and descent procedures.
Within this context, this notebook, along with VPM7* and VPM8 lay a framework for analyzing profiles such as ascents from a spike dive or a saw-tooth profile. While these are not idealized square profiles, they can't be called non-standard either --divers do these and other bizarre profiles all of the time.
*YH denotes the Yount and Hoffman paper cited in VPM6. Equations from YH are cited as (YH #). VPM# refers to the VPMechanics# notebook.  

The Dynamic Critical Volume Hypothesis

The YH dynamic critical volume method constrains ascents by the condition:

' <= α    (YH A1)

Here, the notation   - is changed to , with an addition of the tags j to denote variation of nuclear distributions by compartment.
For this analysis of a single, non-stop decompression profile, the are held constant during ascent and after surfacing. The distributions are, however, certainly time dependent following the dive.

Setting up

On Ascent
The last plot of the VPM1 notebook shows a typical set of converged gradients for a non-stop dive. From the start of the linear, 33 ft/min ascent until surfacing, the gradients are nearly linear in time. Upon surfacing, the gradients decay to the negative inherent unsaturation. In the following discussion, the notations of  VPM7 and VPM8 are used.
The main departure from the YH 's constant ascent gradient will be use of a linear form for(t) that rises from at the start of the ascent to after surfacing at . So, during ascent,

(t) = t +),   

where denotes the magnitude of the supersaturation gradient. This simplifies to:

(t) =   t +) , 0 <= t <

After Surfacing
To keep the focus on the ascent, and also to allow analytic evaluation of the supersaturation integrals, let's retain the YH form for the (t) after surfacing.

(t) = exp(- t), <= t

Here, the YH halftimes H* are expressed as .

Calculating

The critical volume condition becomes:

<=  α

After integrating and adopting the equals sign,

[) - ) ] = α .
With the YH definitions:

~~ ( - ),   Note: there's a typo in the YH paper at this point (YH A9)
λ = ,   (YH A15)

and a little algebra grind, there results:

The Dynamic Critical Volume Equations

=   (anyone want to test-dive the negative root?)

= + -

= -

Discussion

The last terms in the expressions for and add to the original expressions for b and c in the YH paper, and reduce to the YH equations if , = 1, and = 0, which describes saturated Jell-O pretty well.
These new terms can have magnitudes that are comparable to the others, and increase the gradients allowed during ascents from the YH values for equivalent ascent times and pressurization-depressurization schedules.
It should be emphasized that the ascent time is always greater than zero. This model assumes a constant ascent rate, = , so the dynamic critical volume equations are not intended to handle dives with decompression stops. Furthermore, the effects of the use of a non-zero ascent time far out ways the new terms in the gradients.

The following figure plots three different calculations of , using γ = 17.9 dyne/cm, γc = 257 dyne/cm, and = 1 μm. As with previous notebooks, the ZHL16 set of half-times was used for computation.
The gray curves are identical to those shown as the maximum supersaturation gradients in VPM4, calculated using the YH equations 3 and 4, with =0.
The rainbow of continuous curves were calculated using the YH equations with   = and Ascent Rate = 1 atm/min. So, for example, on an ascent from 100 ft, with ΔP ~ 3 atm, = 3 min. Different halftimes are denoted by colors, extending from the red-colored curve for the 4-min half-time to the purple-colored curve for the 635-min half-time
The open circles are the values of calculated using this notebook's with a 1 atm/min ascent rate, in conjunction with the VPM7 form for , which corrects for the 3 atm/min descent rate. Note that as in the diagrams shown in VPM7, the are all inverted prior to being converted to by the dynamic critical volume equations. Here, the open circles lie slightly below the YH curves. So, the combined effects of finite ascent-descent rates, and explicit use of compartment tensions to calculate mitigate the apparent >= b, >= c of the dynamic critical volume equations.

Clearly, with proper ascent time, the original YH equations are sufficient for modeling no-stop nitrox dives made with punctual descents.

Notes

1) As described in VPM4 and VPM5, the above curves are not the same as typical Gradient vs. Pressure or M-value vs. Pressure curves.  In the VPM, hyperbaric M-values are parallel to ambient pressure, and offset by the set of   corresponding to the dive depth ΔP.

2) The range of ΔP shown in the above plots is considerably larger than the accepted safe range for air / nitrox diving. In particular, the 7 - 9 atm range is used only to illustrate the approach to the impermeable range.

3) A slower descent rate would significantly depress the open-circle values for the corresponding to the deeper dive depths.

4) All of the curves converge to the slow-compartment's purple curve for extremely slow ascents or for large ΔP.
    
5) The curves revert to the gray-colored forms for ->0

The updated big picture

The following graph shows the transformation of the PssMin surfaces shown in VPM7 under the dynamic critical volume equations. The same form of the equations that generated the previous graph's solid curves was used for the calculations. The previous graph is a plot of the intersection of the Descent Rate = 1 atm/min plane with these surfaces. Note that the conventional ordering of the supersaturation gradients holds except for very slow descent rates. PssNew is only affected by descent rates less than  ~1atm/min.

A fictitious penalty?

On the one hand, these results produce fast-compartment VPM surfacing M-values that are less than Buhlman's for dives deeper than about 100 ft --which can't be a bad thing. On the other, they seem to encourage rapid ascents. In other words, with   =0, a diver could enjoy the maximal as shown by the gray-colored curves. So, does this formulation effectively penalize divers for conservative ascents?
I don't have a pat answer for this --just a bunch of qualitative objections. Obviously, you can't go from the bottom to the surface instantly, and even a 1-min ascent time significantly reduces the gradients. Also, for dives deeper than 100 ft, you would be constrained by the fast compartment's ~ 3 atm supersaturation limit. Further, it could be argued via the diffusion equation that a rapid ascent causes a rapid change in bubble radius. Then there's the model itself --we ignored the oxygen window to make the surface Pss integral analytic, but if the integral was calculated as in VPM2, would the "risk" be larger and extend further in time? And on and on....

No-stop time limits

The following table delineates the no decompression limits (NDL) found using the methods that produced the open circles in the above figure. In all cases, there was a 1 atm (33 ft) per minute ascent rate.

   
1.0 / 33            834       
1.2 / 40            217
2.0 / 66             40
3.0 / 99             15
4.0 / 132           8
5.0 / 165           4
6.0 / 198           3