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2
On the arbitrariness of
the ZH-L Helium coefficients
Abstract:
During our tests [1] with the highly topical SCUBAPRO / Uwatec mix-gas dive
computer G2 TEK and the SHEARWATER PERDIX, we found divergent
results in Heliox run-times. Thus we set to work to scrutinize the sources of
the Helium coefficients within the ZH-L framework [attachment], which are
needed to calculate decompression obligations.
Introduction: slides # 3 6
Methods: slides # 7 & 8
Results: slides # 9 12
Discussion & Conclusion: slides # 13 & 14
References: slide # 15
Attachment: following slide # 15
3
On the arbitrariness of
the ZH-L Helium coefficients
Introduction (1):
The ZH-L framework (with „ZH“ = Zürich, the town in Switzerland (CH) were
Albert Alois Bühlmann et al. worked), „L“ = linear, since the equation to
calculate a tolerated inertgas overpressure in a theoretical compartment is
a simple, linear equation) [pls. cf. the „attachment“] ) belongs to a group of
deterministic decompression models where the blood perfusion in a
theoretical body tissue (aka „compartment“) dominates the inertgas-uptake
and –release processes for N2 or Helium in solution. Diffusive processes and
a free gas-phase are not considered.
The compartments are earmarked by a half-time (HT, t1/2). For a linear
equation two parameters are needed: an axis interception and a slope.
For e.g. Haldane et al. used 5 compartments with HT from 5 to 75 min for Air,
Workman et al. used 9 with HT 5 240 min both for N2 and Helium;
Schreiner et al. and Müller & Ruff used 16 for each inertgas breathed.
4
On the arbitrariness of
the ZH-L Helium coefficients
Introduction (2):
As the ZH-L framework offers 16 Helium compartments with HT from ca. 1 to
240 min, there are 16 * 3 free parameters to be determined via experiments,
be it with animals or human volunteers.
Bühlmann et al. focused their experiments with Helium on deep & saturation
diving, with at times unpleasant collateral effects for the volunteers (pls. cf.:
https://www.divetable.info/BS_ZH/Q&A.pdf ).
There have been only a few experiments on bounce-/ sprint-dives,
(according to Thomas A. Bühlmann: 455 overall, source:
https://www.divetable.info/BS_ZH/Medical_Research_ZH.pdf) thus the
commercial diving community needs to know, how reliable the thus derived
Helium-coefficients are, or if there are any safety- & security-caveats for
the complete set of ZH-L16A Helium coefficients.
5
On the arbitrariness of
the ZH-L Helium coefficients
Introduction (3):
Basically we use a simple box-profile with a bottom depth of 42 m and a
bottom time of 25 min with regular air as a breathing gas to compare dive
tables, dive computers or decompression algorithms ([1] & [2] with all the
references therein).
This particular profile, when dived on air, is intermediate between recreational
and technical diving; it could be dived for an advanced student on a single
tank with 12 L water-capacity, even if she uses in cold waters a dry-suit.
When low breathing resistance and low narcotic potential due to the lack of
nitrogen is at premium, it could be used as a paradigm for Heliox diving, say
for a bounce dive to clean a sewer pipe or for inspection of a dam outlet.
By using a Heliox mixture with the same inertgas fraction as air, i.e.:
Heliox 21 with 21 % of Oxygen and 79 % of Helium, differences in
performance or computation can be unveiled very quickly.
6
On the arbitrariness of
the ZH-L Helium coefficients
Introduction (4):
The divers body and all of their theroretical compartments are nearly
saturated with nitrogen at ambient pressure, prior to the dive. When she starts
to breathe Heliox from the surface, ICD (isobaric inertgas-counterdiffusion of
the „deep tissue“ type) takes place in her body: Helium diffuses quickly in and
Nitrogen slowly out due to a maximal inertgas gradient, for both gases of ca.
0.8 Bar.
It this mechanism is not properly implemented, being in a dive table or a
dive computer, and / or proprietary modifications of a published ZH-L16x
matrix are used, anomalies in the resulting run-times will leap into view.
By comparing identical dive-profiles with air, a Heliox run-time starts the
decompression phase deeper and overall is substantially longer in the
shallow phases.
7
On the arbitrariness of
the ZH-L Helium coefficients
Methods (1):
During our latest tests with the highly topical Scubapro / UWATEC Galileo G2
TEK [1] we found various discrepancies in the run-times for this particular
Heliox profile. These discrepancies we attributed basically to:
different methods of calculation
a modified set of Helium ZH-L coefficients
as the results for the same profile on air do differ, but not substantially.
We compared these results from the dive computers integrated dive-planners
with well-known, publicly documented, desktop decompression tools [3] & [4].
These two tools are using definitely the original ZH-L16C Helium coefficients
with a very high intrinsic floating point precision along with a comparable
numerical calculation method required to calculate the decompression
obligations.
As well we analyzed the coefficients matrix as such with algebaric methods
[attachment].
8
On the arbitrariness of
the ZH-L Helium coefficients
Methods (2):
Both of the used dive computers claim to use a ZH-L16x perfusion model.
But none of the manufacturers clearly unveils the used set of constants:
neither for N2 nor for Helium. As well other, pivotal constants needed to
calculate a decompression obligation, like the water temperature, the
used water density or the respiratory quotient are not openly documented.
In stark contrast to printed dive tables, the end-user of the electronic
equipment (the diver) is left completely in the dark concerning the
performance and the safety & security of her dive computer.
As the OSTC tool as well does not clearly identify the used water density and
the respiratory quotient, we assumed the following for our own calculations
and as input to the dive computers planning modes:
ambient water temperature: 20 ° C
water density EN 13 319 or „sea water“
respiratory quotient = 1.00
11
the „test dive“ 42 m / 25 min with Heliox 21/79, GF 100 / 100
water density salt / sea resp. EN 13 319 for DIVE
(*) DIVE 3_11 with the reduced Helium-Matrix, pls. cf. [attachment]
(**) DIVE 3_11 with the ZH-L12 Helium coefficients from 1983
Results (3):
stop times 12 m 9 m 6 m 3 m ca. TTS
Computer:
G2 TEK 7 18 26 39 96
Perdix 2 7 15 36 63
Software:
DIVE 3_11 3 8 17 44 75
OSTC
Planner
3.13
2 7 16 45 75
DIVE 3_11 98 (*)
DIVE 3_11
(**) 18 m / 2
15 m / 4
12 m/ 9
13 22 46 101 (**)
12
This new, reduced Helium Matrix has only 6 compartments
with relatively arbitrary a- & b-coeffcients: it recovers all of the
run-times within an error of ca. +/- 20 min. TTS. This yields as well for the
30 m / 120 min bounce dive from [6].
The „test dive“ 42 m / 25 min with Heliox 21/79, GF 100 / 100;
water density EN 13 319; with the reduced Helium-Matrix
pls. cf. [attachment] and the thus resulting run-time:
The # of compartments is put to a minimum of 6 in order to minimize the
degrees of freedom. As well we kept the orginal HT in order to minimize
the algorithmic search efforts.
Results (4):
13
Discussion (1):
The OSTC Planner is now since 2016 in 3.00 BETA 14 phase,
with no further developments, thus we had to rely on our own
asumptions about parameters (pls. cf. methods slide # 8). In the past,
we had compared already 480 Heliox profiles with perfect agreement [5].
The run-times here are also in perfect agreement.
We did run the simulations from [1] again, but with the salt-/seawater settings
for the water density in order to get similar parameters as the software products.
As the G2 TEK and the PERDIX differ by ca. ½ hour, resp. - 10 min / + 20 min.
to the two software run-times, our conjectures (slide # 7) are well-grounded.
Finally, and for the fun of it, we used as well the ZH-L12 Helium coefficients
from 1983, pls. cf. the attachment: this run-time is by far the longest.
As the variability of the Helium-coefficients over time and as well, over the
various systems is maximal, their application to commercial diving
should be excercised only with caution!
14
Discussion (2):
The perfect agreement of the two desktop deco-programmes
should not entrap us of the validity or plausibility of this
run-time: it just tells us, that we, in this club, did make the same errors, or:
different errors with just the same outcome.
As there is a bandwidth of the TTS of ca. 40 min, this by no means implies,
that the two TTS derived from software could be correct and the
others were not! The diverging run-times from slide # 11 have never been
seriously challenged by controlled & reproducible experiments.
A commercial diving operation would run this dive completely different
anyway: with a high pO2 during the last 15 m of decompression.
But this is the topic of another discussion, and, maybe of a to-be
presentation here for RESEARCHGATE; i.e.: a benchmark of
Heliox tables from the commercial, C&R (construction & repair) or military
sector.
15
References (1):
[1] Miri Rosenblat, TAU; Nurit Vered, Technion Haifa; Yael
Eisenstein & Albi Salm, SubMarineConsulting (27.07.2022)
On the reliability of dive computer generated run-times, Part VIII: G2 TEK
DOI: 10.13140/RG.2.2.22374.50247
[2] Miri Rosenblat, TAU; Yael Eisenstein, SubMarineConsulting (28.02.2022)
On the reliability of dive computer generated run-times: Synopsis &
Closure
[3] OSTC Planner, accessed 08 / 2022
[4] Salm, A. (17.01.2022) Recovery of selected ZH-86 air-diving
schedules via a decompression shareware).
[5] Salm, A. (03 / 2021) update on the 2010 CAISSON paper
[6] Salm, A.(10 / 2020 ) ZH-L12: Validation of an old (1982) experimental
Heliox jump dive (30 m, 120 min), DOI: 10.13140/RG.2.2.24608.20482/1
ZH-L 16 Helium Coefficients
ALBI @ www.SMC-de.com
1
The ZH-L16 Helium coefficients
Contents:
Contents
The ZH-L16 Helium coefficients ........................................................................................... 1
Historical perspective ............................................................................................................ 5
A reduced Helium coefficients-matrix ................................................................................... 8
References for this attachment: ............................................................................................ 9
Basis of the following algebraic analysis is the original table from [65], p. 158:
ZH-L 16 Helium Coefficients
ALBI @ www.SMC-de.com
2
The formula to derive the coefficient a from the used Helium compartment halftimes:
a = 2.0 Bar * t ½ He [min] -1/3
(This is the identical formula from p. 129 for N2)
But according to p. 131 this formula is valid for the a-coefficients only for the compartments #
1 – 8, l.c. the printed paradigm for compartment # 8:
a #8 = 2.0 * 29.11-1/3 = 0.6502
But for the compartments # 1 - # 7 it looks like that:
ZH-L 16 Helium Coefficients
ALBI @ www.SMC-de.com
3
a formula – a table
a #1 = 2.0 * 1.51-1/3 = 1.7433 0.0009
a #1b = 2.0 * 1.88-1/3 = 1.6205 0.0016
a #2 = 2.0 * 3.02-1/3 = 1.3837 0.0007
a #3 = 2.0 * 4.72-1/3 = 1.1923 0.0004
a #4 = 2.0 * 6.99-1/3 = 1.0460 0.0002
a #5 = 2.0 * 10.21-1/3 = 0.9219 - 0.0001
a #6 = 2.0 * 14.48-1/3 = 0.8206 0.0001
a #7 = 2.0 * 20.53-1/3 = 0.7304 - 0.0001
From compartment # 9 on this should no longer be used, but instead a factor of 2.0 2.2
with the already tabulated N2-values:
Δ to tabulated values:
a #16 = 2.2 * a#16 N2 = 2.2 * 0.2327 = 0.5119 0
a #15 = 2.0 * a#15 N2 = 2.0 * 0.2523 = 0.5046 - 0.0126
For an analysis of one constant C for the rest of the compartments #14 – 9, we set the a-
coefficients in proportion. This reveals, that the reading of 2.0 on p.131 for compartment #15
is also wrong:
a #15 / a #15 N2 = C = 0.5172 / 0.2523 = 2.0499
a #14 / a #14 N2 = C = 0.5176 / 0.2737 = 1.8911
a #13 / a #13 N2 = C = 0.5181 / 0.2971 = 1.7439
a #12 / a #12 N2 = C = 0.5189 / 0.3223 = 1.6100
a #11 / a #11 N2 = C = 0.5333 / 0.3497 = 1.5250
a #10 / a #10 N2 = C = 0.5545 / 0.3798 = 1.4600
a #09 / a #09 N2 = C = 0.5950 / 0.4187 = 1.4211
For the compartments # 01 – 08 the argument is now an 1.380 – 1.40-times increased
solubility:
a #08 / a #08 N2 = C = 0.6502 / 0.4701 = 1.3831
a #07 / a #07 N2 = C = 0.7305 / 0.5282 = 1.3830
a #06 / a #06 N2 = C = 0.8205 / 0.5933 = 1.3829
a #05 / a #05 N2 = C = 0.9220 / 0.6667 = 1.3829
a #04 / a #04 N2 = C = 1.0458 / 0.7562 = 1.3830
a #03 / a #03 N2 = C = 1.1919 / 0.8618 = 1.3830
a #02 / a #02 N2 = C = 1.3830 / 1.0000 = 1.3830
a #1b / a #1b N2 = C = 1.6189 / 1.1696 = 1.3841
a #01 / a #01 N2 = C = 1.7424 / 1.2599 = 1.3830
Basically Grahams Law should hold for the relation of the halftimes with the factor of 2.65
(p. 98):
[ Excursion
Grahams Law:
The diffusion-velocities of 2 gases relate as the square-root of their respective molecular
weights:
molecular weight Nitrogen (N2): ca. 28
Helium (He) ca. 4
the relation is then: 28 / 4 = 7
and the square-root from 7 is ca. 2.65: √7 = 2.6458
ZH-L 16 Helium Coefficients
ALBI @ www.SMC-de.com
4
Excursion: End ]
An analysis of the halftimes ( t1/2 ) per compartment reveals:
compartiment # relation t1/2 N2 / t1/2 He:
#1: t1/2 N2 / t1/2 He = 4 / 1.51 = 2.6490
#1b: t1/2 N2 / t1/2 He = 5 / 1.88 = 2.6596
#2: t1/2 N2 / t1/2 He = 8 / 3.02 = 2.6490
#3: t1/2 N2 / t1/2 He = 12.5 / 4.72 = 2.6483
#4: t1/2 N2 / t1/2 He = 18.5 / 6.99 = 2.6466
#5: t1/2 N2 / t1/2 He = 27.0 / 10.21 = 2.6445
#6: t1/2 N2 / t1/2 He = 38.3 / 14.48 = 2.6450
#7: t1/2 N2 / t1/2 He = 54.3 / 20.53 = 2.6449
#8: t1/2 N2 / t1/2 He = 77.0 / 29.11 = 2.6451
#9: t1/2 N2 / t1/2 He = 109.0 / 41.20 = 2.6456
#10: t1/2 N2 / t1/2 He = 146.0 / 55.19 = 2.6454
#11: t1/2 N2 / t1/2 He = 187.0 / 70.69 = 2.6454
#12: t1/2 N2 / t1/2 He = 239.0 / 90.34 = 2.6456
#13: t1/2 N2 / t1/2 He = 305.0 / 115.29 = 2.6455
#14: t1/2 N2 / t1/2 He = 390.0 / 147.42 = 2.6455
#15: t1/2 N2 / t1/2 He = 498.0 / 188.24 = 2.6456
#16: t1/2 N2 / t1/2 He = 635.0 / 240.03 = 2.6455
The formula for the derivation of the b-coefficient from the halftimes (p. 129) is only valid for
N2: the majority of the Helium b-values are tumbling out of the blue. The procedure on p. 131
(„etwas reduziert“ reads: reduced a little bit) is not precisely defined. The reduction is ca.
0.08 for the fastests compartments and up to ca. 0.03 for the slowest ones, but with a certain
arbitrariness. This analysis of the reduction from N2 b-value to the He b-value shows a clear
tendency, but no rationale, be it physiologic or purely mathematical:
compart N2 b-value - He b-value:
-ment #
#1: 0.0805
#1b: 0.0808
#2: 0.0767
#3: 0.0695
#4: 0.0602 (resp.: 0.0502)(*)
#5: 0.0544
#6: 0.0477
#7: 0.0414
#8: 0.0357
#9: 0.0335
#10: 0.0319
#11: 0.0322
#12: 0.0330
#13: 0.0355
#14: 0.0373
#15: 0.0385
#16: 0.0386
ZH-L 16 Helium Coefficients
ALBI @ www.SMC-de.com
5
[
(*): for comp. #4, the reduction is 0.0602 with the published #4 N2 b-value from the table
above. But this b-value in itself is not derived from the N2-formula on p. 129:
b = 1.005 – 1 * ( t1/2 N2 [min]) -1/2
i.e.: the #4 N2 b-value should not read 0.7825 but instead:
1.005 – (18.5) -1/2 = 0.7725
And with this the reduction should now read 0.0502
]
Only for the compartments #11 & 16 are some results published: on p. 144 & 145 we find the
two graphs Pamb,tol vs. Pt He. These are the theoretical linear functions and a couple of
„measured results“, the complete raw data are not published. Due to the spread of the x- & y-
axis, for e.g. in Abb. 31 / p. 144 with Δp = 25 Bar and the line-width in combination with the
size of the „dots“ (albeit without any reference to real measurement errors or with displayed
error bars!) we could derive the a-/b-values only with an absolute error of ca.
+/- 2 * 3,3 = ca. 7 %. With this one and the above analysis the Helium a- & b-coefficients for
the medium compartments are reduced to a certain arbitrariness.
Historical perspective
Long before the ZH-L12 Helium coefficients have been published ([4] & [243]) other groups /
researchers have been using their own, genuine sets of parameters. The complete historical
perspective is compiled in [250], chapters 11 – 16, with separate chapters from the historical
POV along with an in-depth description by Hills, B. A. of the then topical thermodynamical
model, the USN, the british, the canadian and the swiss perspective.
Workman et al. used 9 compartments with the same HT, for both N2 & He (Source: Workman
R. D. (1965) NEDU Report 6-65, p. 31 & 32), i.e. without the Graham-factor, but with different
surfacing values (M0 in [fswa]):
ZH-L 16 Helium Coefficients
ALBI @ www.SMC-de.com
6
Schreiner et al. used 15 (Source: Schreiner, H.R., and Kelley, P.L. "A Pragmatic View of
Decompression," Underwater Physiology Proceedings of the Fourth Symposium on
Underwater Physiology, edited by C.J. Lambertsen. Academic Press, New York, (1971) pp.
205-219 (the 16th. could not be used due to a restricted capability of the line-printer
connected to their Burroughs-Computer (Source: R.W Hamilton, 9.th UHMS workshop, p.
92).
Müller & Ruff used a modified Schreiner-Matrix with 15 compartments; the HT are for
N2: 4 416 min and for Helium: 3 139 min. There is as well no Graham-factor:
(Source: Müller & Ruff (1964) Experimentelle & theoretische Untersuchung des
Druckfallproblems, DLR Archiv, Köln)
ZH-L 16 Helium Coefficients
ALBI @ www.SMC-de.com
7
The ZH-L12 set has indeed a history of its own: the first, closed-shop document was the book
from Bühlmann et al. in 1983 [4] and its immediate follow-up as an english version in 1984
[234]. Here we find the following sets of Helium-coefficients:
Source: [4], p. 27, 1983
Source: [234], p. 27, 1984
We found discrepancies between the two sets at the red dots.
ZH-L 16 Helium Coefficients
ALBI @ www.SMC-de.com
8
A reduced Helium coefficients-matrix
In order to demonstrate that relatively arbitrary a- & b-coefficients could be used to recover
run-times in a sound manner, we set up a multi-dimensional search algorithm with a so-
called „Evolution Strategy“, i.e. the local minimum of a multi-dimensional hyper-surface is
found by randomly, simultaneous changed parameters (Source: Rechenberg, Ingo (1973)
Evolutionsstrategie: Optimierung technischer Systeme nach Prinzipien der biologischen
Evolution, Frommann-Holzboog # 15, ISBN 3 7728 0373 3).
In order to keep the search efforts minimal, we decided to keep 6 of the original HT, ie. these
from the original compartments # 1, 2, 5, 8, 11 & 16. Thus we reduced as well the degrees of
freedom, i.e. the simultaneous variation of the parameters to 12 (6 * a + 6 * b).
After a couple of hours of CPU time in our Build-Server Array, we recieved the following
result, with, from left to right:
the compartment # half-time a-coefficient b-coefficient (ignore the λ)
The reduced Helium-Matrix
This matrix is loaded into the service engine of DIVE to simulate the run-times in
question.
ZH-L 16 Helium Coefficients
ALBI @ www.SMC-de.com
9
References for this attachment:
[4] Dekompression - Dekompressionskrankheit, A. A. Bühlmann, Springer, 1983, ISBN 3-
540-12514-0
[65] "Tauchmedizin.", Albert A. Bühlmann, Ernst B. Völlm (Mitarbeiter), P. Nussberger; 5.
edition in 2002, Springer, ISBN 3-540-42979-4
[234] Bühlmann, Albert Alois (1984) Decompression - Decompression Sickness, Springer,
ISBN: 3-540-13308-9, 0-387-13308-9, e-book 978-3-662-02409-6
The „Bühlmann Six-Pack“:
ZH-L 16 Helium Coefficients
ALBI @ www.SMC-de.com
10
[250] Bennett, Peter B., Elliott, David H.(eds.) (1969) The Physiology and Medicine of Diving
and Compressed Air Work, First Edition, Bailliere Tindall and Cassell, London ISBN: -
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