VPMechanics7

Copyright 1999-2001 Eric Maiken

Descent Rate and PssMin in the VPM Permeable Regime

This notebook considers the role of a finite descent rate on setting the minimum allowed supersaturations PssMin for nitrox diving.

With all the equations that follow, maybe it's best to choose an easy-going name for the effect. How about: The Inverse Gradient Bubble Model ?

The results of this analysis are a set of counter-intuitive, upside-down gradients like those discussed in the VPMechanics4 notebook for the impermeable regime. For the entire permeable pressure range, the gradients allowed for the slow compartments are greater than those allowed by the fast compartments.  
In the notebooks VPMechanics8, and VPMechanics9, the effects of ascent rate on PssMin and PssNew are discussed, and it is found that Yount and Hoffman's dynamic critical volume method flips the gradients back over to the conventional sequence.
Regardless of whether this is just a mathematical exercise, the physics is clear. Divers should descend on the highest ppO2 mix, with the slowest diffusing inert gas possible --especially if the descent rate is slow. For example, this would imply that on a trimix dive, you should head down on the richest possible nitrox, and then rapidly descend once switching to your bottom mix. Practically this will be mitigated somewhat by O2 toxicity and narcotic effects.

First, some background material

Start with equation 11 from Yount's Skins of varying permeability paper (S).
2(γc-γ)= -      (S11)
However, rather than setting = 0 as was done in eq. (S12), this notebook  tracks by compartment according to a linear compression and exponential rate equations. The compression is assumed to be within the permeable regime. As noted by Erik Baker, the onset of impermeability may depend on compression rate and vary by compartment. Nonetheless, the non restrictive delineation of = 9 ata is used to set a rough boundary for the permeable regime in this analysis.

Assume a linear rate of descent: =+ t. Then,

=   ∂ t

Now, with the change in notation: -> , denoting the time dependence of the tension in the j'th compartment on descent, the set of tensions are:
= + (t - )  + ( ) exp(- t), 0 <= t <=
See the appendix at the bottom for details on the development of this equation.
The differentials for the compartment tensions are:

-> = - ( ) exp(- t) ] ∂t ,   t <=

Solve S11 for descent with = compartment tensions and γc a constant

Setting  δ =0, treating γc as a constant, denoting the descent time by , and assuming the same initial distribution of nuclei in all compartments, equation S11 can be expressed as:

2(γc-γ) =  [ (1-exp(- t) ) - 1] dt
Note that if t is set to 0, and is set to r in the integrands, this reduces to Yount's form, with = ΔP = -.
Integrating and setting =0 produces the result:

2(γc-γ)( - ) = (1 - ) + [1 - exp(- ) ]

Solve S11 for descent with = compartment tensions and γc  = α

By eq. (S10), it may be appropriate to set: γc = α , where α is constant.
Then: γc = α= α[ + (t - )  + exp(- t) ],
and (S11) becomes an integral that cannot be easily evaluated in closed form:

2 ( - ) =  

Apply the solutions for the to determine the minimum allowed supersaturation gradients for ascent

Let's simplify the situation by determining the set of PssMin for an instantaneous ascent. So,  eq. (S11) is applied again with γc = constant, = 0, using the threshold Laplace conditions: -.   is the set of compartment tensions at the end of the dive/ beginning of ascent, and is the minimum allowed ambient hydrostatic pressure --ie: the first stop required for the j'th compartment.
In analogous forms to equation (S20), the sets of PssMin are:

Constant γc  descent:
= + ,
with = ΔP(1 - ) + (1 - exp(- ) ).

Variable γc during descent:

=

What's the difference?

For both constant  γc and variable γc, there are dispersions in the PssMin. Counter to the conventional ordering of gradients, the slower compartments are predicted to resist higher supersaturation pressures than the faster compartments. This is because the equilibrium nuclei in fast compartments are calculated to be larger than those in slow compartments. Physically, this results from the exposure of nuclei in fast compartments to larger dissolved gas tensions during descent than the tensions surrounding nuclei in slow compartments.
The figure below shows two sets of PssMin, with the variable γc set displayed as the rainbow extending down from the purple-colored 635 min compartment, to the red-colored  8 min compartment. Note that solutions for only 15 of the 16 ZHL-16 half-times were calculated --I couldn't do the  variable γc integral numerically for the fastest 4-min half-time. The gray set are the results for constant γc, ordered from slow down to fast. The curves were calculated for a relatively slow descent rate of 1 atm/min. So, for example, it required 8 min to reach the ΔP = 8 atm mark.
Note that the uppermost (slowest compartment) gray curve is identical to the conventional single PssMin curve that applies to all compartments as defined in VPMechanics1 and VPMechanics4. So, here, all the faster compartments fan out from the static results for the slowest compartment.  The descent was calculated for air. If helium were used, the dispersion would be more dramatic. The linear pressure dependence of the variable γc was set by the parameter α, which was chosen to be equal to the constant γc = 257 dyne/cm at one atmosphere. This results in the difference in slope between the slowest compartments, which are tracked by the uppermost gray, and the purple-colored curves.

The big picture

The global behavior of the set of PssMin is tracked vs. descent rate and change in pressure ΔP in the following figure for constant γc = 257 dyne/cm. The conventional VPM PssMin is described by a single plane for all compartments, which is virtually identical to the uppermost purple-hatched surface, representing the slowest, 635-min half-time compartment.
This limit is reached in the present analysis as seen by the convergence of the surfaces at rapid descent rates.

Appendix: Calculating for a linear compression

Apply Fick's law:
, for each compartment j.
Rewrite as the 1st order nonhomogeneous equation:

With:
initial conditions
      and
      = ( - ),

the time dependence of the alveolar pressure:
     = + ) = + t (t) ,   (t) = > 0,
and the notation:
= +, for compartments j and alveoli A, to define δ = -~~ 0.

The solution set is:

= + (t - )  + ( ) exp(- t), 0 <= t <=