Question regarding Baker's Decolessons

[EFX]

I believe Psb means pressure at sea level base (or ambient). Using the Workman + GF formula from above:

Workman + GF: M = D * (dM * GF - GF + 1) + (Psb + GF * (M0 - Psb))

Using a depth of 0 for the surface and a GF of 0 the formula reduces to:

M = Psb

Recall from the graphs that a GF of 0 is a TC pressure from a point on the ambient pressure line. A GF of 1 (100%) is a TC pressure from a point on the m-value line. For the ascent to surface, setting Psb to the ambient sea level pressure of 0 and a GFHi of 75%, the Workman GF formula reduces to M = 0.75 * Mo, just as you surmised. Now that we understand the equations let's get back to the question:

Why is Baker's ascent to surface m-value equal to 91.2% of Mo and not 75%?

 

[EFX]

Just to summarize, in Baker's paper "Decolessons" he presents a methodology for a decompression schedule and a sample program to implement it. He then includes a sample output of his program: a dive to 90 m for 20 minutes. The output shows several columns of data, one of those being the stop depth, stop time, and the segment's GF. Interspersed with the deco stop rows he inserts rows of ascent time showing the ascend to depth, rate, and %m-value.

His calculation for m-value uses a formula from another paper called "Understanding M-values". The formula is:

Ptol = (Pamb / b) + a

This is a Buhlmann-style formula he uses to calculate the m-value when the a and b coefficients are known. Workman style formulas can be used as I and dmaziuk have shown above. To get %m-value Baker takes the inert gas load of the leading tissue compartment (CTC) and divides it by the m-value (the Ptol) and then multiplies by 100 to get percent:

%Mv = Pctc / Ptol * 100

The %Mv is what Baker records in his sample output. So, dmaziuk and I have been exploring (through the equations) what does Ptol mean and how is it calculated and can we resolve the discrepancy between the m-value of 91.2% on surfacing and the GFHi of 75% also recorded upon surfacing. But, there is another factor we need to look at: what I call the Pctc or total inert gas pressure of the controlling tissue compartment. In order to give a result equal to 75 the value Baker calculates for it is too high.

I used Baker's formulas above in my spreadsheet and came up with different numbers for my column %AoM from my original calculation of Pctc / Mo (Mo is the surfacing m-value listed for each tissue compartment). Interestingly, for the surfacing segment both of our approaches came up with the same result. Not surprisingly, this is what we expect because the m-values (Mo) for surfacing are exactly what the equations will give as shown above.

 

[dmaziuk]

If Psb is pressure at sea level, it can't be 0. We'd have a bit of problem with life on Earth and everything if it was. If it's a 1 (bar), for the TC whose M0 is 3.24 at GF 1: 1+ 0.75 * ( 3.24 - 1 ) = 2.68. I.e. M0 at GF 0.75 is 2.68, that is about 85% of 3.24, not 75%.

Unless I'm mistaken, forgot to carry the one, etc., etc.

 

[EFX]

You are confusing gauge pressure with absolute pressure.

 

[dmaziuk]

I don't: he does not spell out in what it is so for all I know it could be partially serrated blades or pimps sitting barefoot. If it's pressure sea-level barometric, barometers don't read 0 IME.

 

[EFX]

You're right. Baker didn't explain Workman's definition in his paper "Decolessons". But he did explain it in another paper called "Understanding M-values". Here is a quote from that paper:

"Workman expressed his M-values in the slope-intercept form of a linear equation (see Figure 1). His surfacing value was designated Mo [pronounced "M naught"]. This was the intercept value in the linear equation at zero depth pressure (gauge) at sea level."

For those of you following this thread who might be wondering what the difference is between gauge and absolute pressure I offer this illustration:

In the US, tanks must be visually inspected every year. This involves bleeding down the tank, removing the valve, inserting a lighted scope and inspecting for signs of rust or defects on the inside tank walls. What is the air pressure inside the tank during the inspection? In imperial units it is 14.7 psia at sea level. At the completion of the inspection the valve is installed. Before we fill the tank let's put a regulator on it with a SPG. What does the SPG read (let's assume for arguments sake it is very accurate)? It reads 0 psig. This is gauge pressure. The absolute pressure inside the tank is still 14.7 psia.

In imperial units we denote the difference by specifying psig (gauge) or psia (absolute). Does metric bar units have a difference in designation?

 

[EFX]

Just trying to get a grip on the problem I ran some numbers using Baker's formulas and here is what I got:

Key (all pressures are absolute):

Ptol = Pressure, tolerated in tissue compartment (TC)
p_gas = pressure of inert gas in TC after ascent to surface
GF = gradient factor
CTC = controlling TC
Mo = surfacing m-value
msw = meters of sea water (pressure)
a = a coefficient
b = b coefficient
s = slope

Dive: Air, Salt, 0 Alt, 30.5 m, 30 min, des = 18.2 mpm, asc = 9.1 mpm
3 m stops, 3 m last stop, Buhlmann ZH-L16C

CTC and pressures were taken from the Dive spreadsheet.
Buhlmann equation for tolerated inert gas at ambient pressure:
Ptol = (p_gas - GF * a) / (GF / b - GF + 1)

GF: 100/100, CTC = 5, p_gas = 18.11 msw
GF: 75/75, CTC = 6, p_gas = 14.97 msw

TC5: a = 6.667, b = 1/s = 1/1.2306 = 0.8126, Mo = 18.5 msw
TC6: a = 5.6, b = 1/s = 1/1.1857 = 0.8434, Mo = 16.9 msw

Ptol(100) = (18.11 - 1 * 6.667) / (1 / 0.8126 - 1 + 1) = 9.2986 msw
Ptol(75) = (14.97 - 0.75 * 5.6) / (0.75 / 0.8434 - 0.75 + 1) = 9.4535 msw

Ptol(75) / Ptol(100) = 9.4535 / 9.2986 = 1.017 = 102%
Ptol(100) / Ptol(75) = 9.2986 / 9.4535 = 0.984 = 98%

GF: 100/100 TC5: p_gas / Mo = 18.11 / 18.5 = 98%
GF: 75/75 TC6: p_gas / Mo = 14.97 / 16.9 = 89%

p_gas(75) / p_gas(100) = 14.97 / 18.11 = 83%

 

[Storker]

In imperial units we denote the difference by specifying psig (gauge) or psia (absolute). Does metric bar units have a difference in designation?

Yep, we have barG and barA. At least in engineering.

 

[dmaziuk]

OK, I got a fresh supply of round tuits:

Workman to Buhlmann: M0 = a + Pamb at surface level / b
With gradient factors: replace a with GF*a and b: with 1/(GF/b - GF + 1)
- M0 = GF*a + Pamb*(GF/b - GF +1)
Assuming Pamb at sea level is 1:
- M0 = a + 1/b
and with GF .75:
- M0 = 0.75 * a + 0.75/b + 0.25

E.g. nitrogen numbers for 5-minute ZH-L16B compartment: 1.2599 and 0.505 resp:
- M0 = 1.2599 + 1/0.505 = 3.24
- M0/.75 = 0.75*1.2599 + 0.75/0.505 + 0.25 = 2.68

2.68/0.0324 = 82.7 i.e. M0 @ GF 75 is ~83% of M0 @ GF 1.

I do have just enough ADD to always forget to carry the minus, or the one, etc. so ICBW and all that.

 

[EFX]

I have a question regarding the output of a decompression program written by Erik Baker, PE in the paper "Decolessons". There are two columns: Max %M-value and Gradient Factor (GF). The dive uses a GF of 30/75 and the output clearly shows the gradual increase of GF from .30 to .75 upon surfacing. Under the Max %M-value column the value upon surfacing is 91.2%. My question is: why isn't the surfacing %M-value equal or less than the GFHi of .75 or 75%?

I finally found an answer to this question which came as a "light bulb going on moment" in another thread. The Max %M-value is NOT another way of expressing gradient factor, it is simply the ratio of the total inert gas of the leading compartment to the m-value at the current depth. For more insight look at figure 3 of baker's paper "Understanding M-values" attached below. Especially revealing is the inset box in the figure. What Baker shows as Max %M-value in the output of his program is the quantity %M-value in the inset. What everyone is interested in however is what is called % M-value gradient in the inset and is described as the ratio of inert gas pressure gradient to the M-value gradient expressed as a percentage. The graph depicts these two quantities. The base of these two quantities is also different: zero absolute inert gas pressure for the %M-value versus ambient inert gas pressure for the %M-value gradient.