Bühlmann ZH-L16 and M-values
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fsardone
Solo Diver
Your response is fine for those who understand this stuff. The OP is trying to understand it, he not there yet. Why don't you help him rather than making fun of my gentle rhetorical attempts to help?A table with all deltaM values (they differ by tissue) is in the referenced file you posted above: Table 1
Just saying.
Buhlmann's a and b coefficients were, according to Powell, calculated as
a = 2 * t ^ -1/3
b = 1.005 - t ^ -1/2
("A" set). There is no explanation, in Baker or elsewhere, that I could find, of which of those 4 numbers are "tolerated surplus volumes" and which: "gas solubilities".
a = 2 * t ^ -1/3
b = 1.005 - t ^ -1/2
("A" set). There is no explanation, in Baker or elsewhere, that I could find, of which of those 4 numbers are "tolerated surplus volumes" and which: "gas solubilities".
OP
David Novo
Contributor
One of your replies was one of the few that adressed the question. I was not asking about the definition of M-value...
I understand what was discussed here regarding the linear equations of Workman and Buhlmann. What I did not get is a clear distinction between M-Value (max partial pressure that a "tissue" can withstand at a given absolute pressure / depth before bubbles being formed) vs. "tolerated surplus volumes".
Thank you for the explanation but it is off-topic.
I am just trying to understand the concept of "tolerated surplus volumes", no need for a mathematical definition
EFX
Contributor
David Novo, I'll attempt to answer your question.
When we ongas, inert gas flows from the lungs to the blood and finally to the tissues. As already described the flow of gas is from a higher to a lower pressure. The Buhlmann model assumes a parallel connection between the source and the tissue (as opposed to a series connection, i.e. tissue to tissue). If we assume no leakage out of the tissue (because it's parallel and not series) during ongassing then the flow of inert gas into that tissue will fill up that volume. Based on this explanation the phrase "tolerated surplus volume" describes an excess amount of gas in the tissue relative to what the volume would be at a lesser depth (pressure). Assuming the tissue can't expand (e.g. bone) then the pressure must increase and that pressure (volume) can be tolerated without DCS if it stays under the m-value.
How were the m-values calculated? From the paper "Neo-Haldanian and bubble models, Buhlmann, bubble grow and bubble dynamics" (see the attached file) on page 7 it says: "For his ZH-L16A algorithm Bühlmann chose to divide the body into 16 compartments and give them a range of T half ‘s, from 4 (or 5) minutes to several hours, 635 (or 640) minutes. The halftimes are arbitrarily chosen, but such that they increase with about the same factor". Buhlmann expressed the m-values in terms of a and b coefficients. In terms of the tissue compartment half-times (ht) these a and b coefficients are:
a = 2ht^(-1/3)
b = 1.005 - ht^(-1/2)
a will be in atmospheres, b is dimensionless.
Now that we have the a and b coefficients we can calculate the m-values based on depth from:
Ptol = (Pamb / b) + a
where: Ptol = the m-value (i.e. tolerated pressure) at depth Pamb. Pamb (pressure ambient) is the pressure at the assumed depth. Note: all values to be absolute pressures. See the paper "understanding_m-values.pdf" for the relevant equations. Powell in "Deco for Divers" also lists some of the equations in the appendix.
When we ongas, inert gas flows from the lungs to the blood and finally to the tissues. As already described the flow of gas is from a higher to a lower pressure. The Buhlmann model assumes a parallel connection between the source and the tissue (as opposed to a series connection, i.e. tissue to tissue). If we assume no leakage out of the tissue (because it's parallel and not series) during ongassing then the flow of inert gas into that tissue will fill up that volume. Based on this explanation the phrase "tolerated surplus volume" describes an excess amount of gas in the tissue relative to what the volume would be at a lesser depth (pressure). Assuming the tissue can't expand (e.g. bone) then the pressure must increase and that pressure (volume) can be tolerated without DCS if it stays under the m-value.
How were the m-values calculated? From the paper "Neo-Haldanian and bubble models, Buhlmann, bubble grow and bubble dynamics" (see the attached file) on page 7 it says: "For his ZH-L16A algorithm Bühlmann chose to divide the body into 16 compartments and give them a range of T half ‘s, from 4 (or 5) minutes to several hours, 635 (or 640) minutes. The halftimes are arbitrarily chosen, but such that they increase with about the same factor". Buhlmann expressed the m-values in terms of a and b coefficients. In terms of the tissue compartment half-times (ht) these a and b coefficients are:
a = 2ht^(-1/3)
b = 1.005 - ht^(-1/2)
a will be in atmospheres, b is dimensionless.
Now that we have the a and b coefficients we can calculate the m-values based on depth from:
Ptol = (Pamb / b) + a
where: Ptol = the m-value (i.e. tolerated pressure) at depth Pamb. Pamb (pressure ambient) is the pressure at the assumed depth. Note: all values to be absolute pressures. See the paper "understanding_m-values.pdf" for the relevant equations. Powell in "Deco for Divers" also lists some of the equations in the appendix.
EFX
Contributor
When deriving the a and b coefficients from the half-times (ht) the a coefficient will be in atmospheres. The b coefficient is dimensionless. For an example we can refer to Erik Baker's paper "Understanding M-values". From the chart of nitrogen ht's and Mo (surfacing m-values) in fsw (feet of sea water) let's calculate the Mo for TC10. TC10's ht = 146 minutes and the Mo = 48.2 fsw under the A column.
a = 2ht^(-1/3) = 2(146)^(-1/3) = 0.3798 atm = 33 fsw/atm x 0.3798 atm = 12.5 fsw
b = 1.005 - ht^(-1/2) = 1.005 - 146^(-1/2) = 0.9222
Ptol = Pamb / b + a = 33 / 0.9222 + 12.5 = 48.2 fsw
The calculated Ptol matches the Mo in the chart. Remember that all values must be absolute so the pressure at sea level is not 0 but 1 atm or 33 fsw or 10 msw (meters of sea water). You can calculate the m-value for any depth by calculating the equivalent pressure in fsw at the selected depth. The m-value at 33 ft of depth for TC10 will be:
First, calculate the pressure at 33 ft in absolute units: 33 + 33 = 66 fsw. Now we can calculate the m-value:
Ptol = Pamb / b + a = 66 / 0.9222 + 12.5 = 84.1 fsw
What this means is when we ascend from depth to 33 ft if TC10's pressure is equal to 84.1 fsw then we're at critical supersaturation and we must stop to allow TC10 to offgas to a safe level before continuing the ascent. This m-value is for a GFHi of 100%. To get the m-value at a GFHi of x% multiply the Ptol by 0.x.
a = 2ht^(-1/3) = 2(146)^(-1/3) = 0.3798 atm = 33 fsw/atm x 0.3798 atm = 12.5 fsw
b = 1.005 - ht^(-1/2) = 1.005 - 146^(-1/2) = 0.9222
Ptol = Pamb / b + a = 33 / 0.9222 + 12.5 = 48.2 fsw
The calculated Ptol matches the Mo in the chart. Remember that all values must be absolute so the pressure at sea level is not 0 but 1 atm or 33 fsw or 10 msw (meters of sea water). You can calculate the m-value for any depth by calculating the equivalent pressure in fsw at the selected depth. The m-value at 33 ft of depth for TC10 will be:
First, calculate the pressure at 33 ft in absolute units: 33 + 33 = 66 fsw. Now we can calculate the m-value:
Ptol = Pamb / b + a = 66 / 0.9222 + 12.5 = 84.1 fsw
What this means is when we ascend from depth to 33 ft if TC10's pressure is equal to 84.1 fsw then we're at critical supersaturation and we must stop to allow TC10 to offgas to a safe level before continuing the ascent. This m-value is for a GFHi of 100%. To get the m-value at a GFHi of x% multiply the Ptol by 0.x.
I am just trying to understand the concept of "tolerated surplus volumes", no need for a mathematical definition
I think "tolerated surplus volumes" is misnomer: the model does not do volumes. I have a feeling Powell didn't read Tauchmedizin either and was just repeating somebody else's handwaving. The best I can figure from various sources, incl. the PDF posted by @EFX, is that Buhlmann did a bit of curve fitting and came up with the mathematical fudge that fits the empirical data. (And whose result had to be later adjusted because it turned out to fit on the "bent" side.) "Volumes" had never been a part of it, it's always ratios and gradients and deltas.
EFX
Contributor
Maybe. I find it hard to believe that Powell misunderstood Buhlmann or he repeated what someone else said without checking the source. We need to look at the 1984 version which was translated into English (I'll search for it later). Buhlmann died in 1994 and the latest version of his book came out in 1995 in German. There is supposed to be a translation of this book in English. Was it published? I couldn't find it. Buhlmann based his algorithm on the results of his experiments. Curve fitting is what modelers do to come up with a mathematical equation that best fits the data. The fact he had to tweak the coefficients of his model based on later experiments shows good judgement in my view. Sometimes approximations are necessary when the model which fits the data much better becomes to complicated to employ.
I'd like to address a subtle point in a previous post of mine regarding the calculations. Take a look at these two statements:
33 fsw != 33 ft of sea water
10 msw != 10 m of sea water
The symbol != means not equal in the C/C++ programming language. You might be thinking "wait a minute" 33 does equal 33 and 10 does equal 10. Yes, that's true. However, fsw does not equal ft of sea water and msw does not equal m of sea water. Why? Well, fsw and msw are units of pressure and ft of sea water and m of sea water are units of distance. To elaborate on this point here is what I wrote in my Excel spreadsheet (ss) on the dive help sheet under the heading "Units of Pressure":
"All pressure values on the ss are displayed as absolute pressures in feet or meters of salt or fresh water rather than the typical units of psi (imperial) or Kpa (metric). This seems odd at first because a distance unit (feet or meters) is used to define a pressure unit. What seems even stranger is that the gauge pressure is equal to the depth in feet or meters. To understand why this is true consider this relationship: 33 fsw/33 ft (or 10 msw/10m). We can describe it this way: there is a pressure of 33 fsw which is equivalent to a pressure exerted by 33 ft (depth) of water. The spreadsheet converts depth to a pressure in order to calculate insP, the inspired inert gas pressure. For example, to convert 80 ft of depth to its equivalent gauge pressure (P) in fsw we could write: P = 80 ft x 33 fsw/33 ft. As a sanity check on the math the ft divide out leaving fsw which is what we want. The really interesting part is that the 33 divides out giving us an answer that is actually the depth but in pressure units of fsw. This is convenient because it eliminates the need to use cumbersome unit conversions in the formulas. Absolute pressures are obtained by adding the pressure of the atmosphere at the surface to the pressure exerted by the water (which is gauge pressure)."
I'd like to address a subtle point in a previous post of mine regarding the calculations. Take a look at these two statements:
33 fsw != 33 ft of sea water
10 msw != 10 m of sea water
The symbol != means not equal in the C/C++ programming language. You might be thinking "wait a minute" 33 does equal 33 and 10 does equal 10. Yes, that's true. However, fsw does not equal ft of sea water and msw does not equal m of sea water. Why? Well, fsw and msw are units of pressure and ft of sea water and m of sea water are units of distance. To elaborate on this point here is what I wrote in my Excel spreadsheet (ss) on the dive help sheet under the heading "Units of Pressure":
"All pressure values on the ss are displayed as absolute pressures in feet or meters of salt or fresh water rather than the typical units of psi (imperial) or Kpa (metric). This seems odd at first because a distance unit (feet or meters) is used to define a pressure unit. What seems even stranger is that the gauge pressure is equal to the depth in feet or meters. To understand why this is true consider this relationship: 33 fsw/33 ft (or 10 msw/10m). We can describe it this way: there is a pressure of 33 fsw which is equivalent to a pressure exerted by 33 ft (depth) of water. The spreadsheet converts depth to a pressure in order to calculate insP, the inspired inert gas pressure. For example, to convert 80 ft of depth to its equivalent gauge pressure (P) in fsw we could write: P = 80 ft x 33 fsw/33 ft. As a sanity check on the math the ft divide out leaving fsw which is what we want. The really interesting part is that the 33 divides out giving us an answer that is actually the depth but in pressure units of fsw. This is convenient because it eliminates the need to use cumbersome unit conversions in the formulas. Absolute pressures are obtained by adding the pressure of the atmosphere at the surface to the pressure exerted by the water (which is gauge pressure)."
fsardone
Solo Diver
Tursiops you ask:
Immediately after postin a file containing the deltaM values for all compartments for several deco models
I say that you just did it
And you scold me to be more rhetorical than you did?
Kettle and pot come to mind but I am not a native speaker and some subleties of your language escape me
Immediately after postin a file containing the deltaM values for all compartments for several deco models
A table with all deltaM values (they differ by tissue) is in the referenced file you posted above: Table 1
Just saying.
I say that you just did it
And you scold me to be more rhetorical than you did?
Kettle and pot come to mind but I am not a native speaker and some subleties of your language escape me
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