VPMechanics8

Copyright 1999-2001 Eric Maiken

Ascent Rate and PssMin in the VPM Permeable Regime

This notebook considers the roles of finite ascent and descent rates in determining the allowed supersaturations for nitrox diving.

Unfortunately, this is all math with no graphics --I haven't coded all these equations....

First, some background material

This section follows the development of VPMechanics7, and determines the by compartment assuming a linear rate of ascent: =- t. So,

=  ∂ t

The notation: --> , denotes the time dependence of the tension in the j'th compartment on ascent.
= - (t - )  +  ( ) exp(- t) , 0 <= t <= ,
with the time from the final dive stage to the first decompression stop (or surface). This equation is developed in the appendix at the bottom.
The differentials for the compartment tensions are:

--> = - ( ) exp(- t) ] ∂t ,   0 <= t <=

Solve S11 for ascent with = compartment tensions and γc as a constant

For constant γc, and assuming that:
i) the initial distribution of nuclei across compartments is determined by the set of radii calculated in VPMechanics7
ii) eq. (S16), which codifies the experimental observation that the nuclear radii don't change during pressurization, is valid
equation (S11) can be expressed as:

2(γc-γ) = [ - ( ) exp(- t) ] dt
Integrating  yields:

2(γc-γ)( - ) = -(1 - ) (1 - exp(- ) )

Sove S11 for ascent with = compartment tensions and γc  = α

Following the above two assumptions, and eq. (S10), which sets: γc = α , with α a constant.
γc = α= α[ - (t - )  +  ( ) exp(- t) ].
(S11) then becomes an integral that cannot be easily evaluated in closed form:

2 ( - )  =  

Put it all together (almost) with constant γc

Here, VPMechanics7's assumption of  an instantaneous ascent is partially relaxed. While the ascent is linear, γc is held constant.
The ascent time is the solution to the implicit equation:

()(+ 1)exp(- ) - [(-1) + 2 (1 - ) + (-1)(- + ) = 0

Appendix: Calculating for a linear decompression

Apply Fick's law:
,
with the initial ascent conditions setting the alveolar pressures to the compartment tensions at the end of the dive
     , and
     = ( - ),
the time dependence of the alveolar pressure:
     = + ) = - t (t) , = > 0
and
= +, for alveoli A.

The solutions are:

= - (t - )  +  ( ) exp(- t) , 0 <= t <= ,