From: mayne@phi.cs.fsu.edu (William Mayne)
Subject: Re: Information about advanced deep diving course
Reply-To: mayne@cs.fsu.edu
Organization: Florida State University Computer Science Department
Date: Fri, 2 Oct 92 17:21:34 GMT

In article <qi27iqg@twilight.wpd.sgi.com> story@sgi.com (David (Duis) Story) writes:
>In article <1992Sep21.141626.22989@mailer.cc.fsu.edu> mayne@cs.fsu.edu writes:
>>In article <q3mkbik@twilight.wpd.sgi.com> story@sgi.com (David Story) writes:
>>>
>>>This [DSAT] algorithm does not allow decompression in the classic Haldanean
>>>sense.  It has no delta-M values (values used to increase the
>>>overpressure allowed at depth, commonly used in Haldanean
>>>decompression models.)  This is not necessarily bad: the delta-M
>>>values can be taken to be zero, which makes it extra conservative.
>>>However, I'm not sure what the decompression computers do, so I can't
>>>say.
>>
>>Either I am confused or this is not correct. If delta-M is taken
>>to be zero then the nitrogen allowed in a compartment is constant,
>>regardless of depth. 
>
>Correct.
>
>>In that case [deltaM=0] the model would say that it is
>>okay to surface or you are already exceeding the ceiling. 
>
>I don't think this is the right conclusion.

Note that if you don't permit the amount of nitrogen in a compartment
to increase with depth you are not keeping ratio of that amount to
saturation (or to ambient pressure, which is simpler) constant. Since
you acknowledge that deltaM=0 does keep the amount of nitrogen allowed
constant I have trouble understanding why you think that holds the
supersaturation ratio constant or that you can calculate staged
decompressions with deltaM=0. That, in a nut shell, is the gist of all
of my argument which follows. I admit that I've only read about this and
I have done the algebra myself. I respect your knowledge, but either one
of us is mistaken or you haven't really been saying what I've been
understanding. Maybe you can help figure out which.

>  Remember that M-values
>are just a convenient notation for an overpressure ratio.
>[Explanation and example deleted.]

This depends upon what you mean by "convenient notation". The M values
can be used to compute the overpressure (or supersaturation) ratio,
but they aren't the same thing. Ratios are not actually used in the
modern model. For historical reasons the supersaturation ratio in some
explanations is given as the ratio of nitrogen in the compartment
to saturation at the same depth, a concept which only works if the
fraction of nitrogen in the breathing mix is constant. (Note that
as the fraction of nitrogen and therefore the partial pressure at
saturation goes to zero so does P_sat and hence the supersaturation
ratio goes to infinity!)

Haldane was working only with air and did originally assume constant
supersaturation ratios. It is interesting to note that sticking with
the assumptions of constant supersaturation ratios and breathing air it
isn't even necessary to consider the fraction of N_2 in air. Everthing
would work just the same if you assumed 100% N_2, which would sure
simplify the math. However that won't work with M values.

An advantage of the actual formulas based on the amount of N_2
tolerated as a function of ambient pressure is that changes in the
breathing mix can be handled as long as you stick to N_2-O_2 mixes.
Again Haldane and explanations based on ratios rather than the amount
of tolerated inert gas in tissues were considering only air so saturation
is .79 times ambient pressure. Hence what Haldane calls a supersaturation
ratio of 2.0 is actually a ratio of 1.58 to ambient pressure.

The supersaturation ratio permitted at the surface is M_0/(.79*33),
using the old style. Check the values shown in Hudgins' paper on the Edge
which you first called my attention to a long time ago. Naturally you
could start with the ratio you wanted and calculate what M_0 should be
just as well.

The meaning of the M value M_d is the partial pressure of N_2 permitted at
depth d and the function of deltaM is to adjust M_d, i.e. M_d=M_0+d*deltaM.
But the supersaturation ratio is clearly M_d/(.79*(d+33)). Note that both
saturation and M_d are linear functions of d and that delta_M is the
slope of the M_d line. The slope of the saturation line (the denominator)
is .79. If deltaM=0 then M is horizontal and the ratio cannot be constant.
In fact the line for M value will intersect the saturation line before long
(as soon as ambient pressure of N_2 reaches M_O) and then be below it,
which would mean you are really in trouble. Hence deltaM must be at least
.79 and in fact it is more than 1.0 for all compartments. For the
supersaturation ratio to constant we need

M_0+d*deltaM     M_0
------------ = --------
.79(d+33)       .79*33

for all d other than -33. (But d=-33 represents a vacuum and we aren't
concerned about that.) Solving this gives the true value for deltaM which
keeps the supersaturation ratio constant:

deltaM = M_0 / 33.

It is actually possible to arrive at this intuitively or by inspection.
If we double the pressure by going to 33 feet we would double the amount
of N_2 permitted if the compartment to maintain the same ratio. Any smaller
value of deltaM means the supersaturation ratio decreases with depth, as is
actually the case. Note that leaving out the .79 fraction of nitrogen
makes no difference. All it does is scale the ratios. This too should be
obvious once you understand that the problem concerns the ratio of two
linear functions. It simplifies the equations, but I left it in because
I didn't want you to think I was pulling the neoprene over your eyes.

Bill Mayne

P.S. It is also interesting to note that the formulas of the Buhlmann
model are based on exactly the same theory. They just represent the
problem as finding the ambient pressure a compartment will tolerate
given any pressure of N_2 in the compartment, rather than starting
with ambient pressure (or, equivalently, depth) and finding the
biggest pressure in the compartment which will tolerate that. Either
can be transformed into the other by simple algebra. For some
calculations it is handy to derive Buhlmann style equations based
on the M and deltaM values, with units converted from bar to fsw.
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