Haldane gave every tissue the same rule: during the ascent, the gas it holds may reach about twice the surrounding pressure, and no more. Schedules were built on that single ratio for about 50 years. As navy divers went deeper and stayed longer, the ratio proved too coarse. Fast tissues could tolerate more than it allowed, slow tissues less, and the tolerated amount appeared to change with depth.
In 1965 Robert Workman, at the US Navy Experimental Diving Unit, replaced the ratio with a separate limit for each tissue that changes with depth. He called it the M-value, M for maximum. It uses 2 numbers per tissue, and every dissolved-gas model since has kept the same form.
- Ambient pressure
- Haldane 1908
- Workman 1965
Haldane’s ratio on a pressure graph
The horizontal axis is the surrounding (ambient) water pressure. The vertical axis is the nitrogen pressure in a tissue. The grey diagonal is a tissue holding exactly the ambient pressure.
Any point above the diagonal is supersaturation: the tissue holds more gas than the ambient pressure. Haldane’s 2 to 1, counted in nitrogen, allows a tissue to reach 1.58 times the ambient pressure. On the graph this is one straight line through zero, the same proportion for every tissue at every depth.
Two numbers per tissue
Workman defined each tissue’s limit with 2 numbers. M0 is the most nitrogen the tissue may hold at the surface. ΔM is the rate at which that limit increases with depth.
His 20-minute tissue may surface holding about 2.2 bar. Its ΔM is 1.5, so for every 10 m of depth (1 bar more ambient pressure) it may hold 1.5 bar more. The limit is a straight line, and it does not have to pass through zero.
A line for every tissue
Workman used 9 nitrogen compartments, with half-times from 5 to 240 minutes, and gave each its own M0 and ΔM.
Fast tissues have high, steep lines: they may hold a large excess of gas, and a larger excess at depth. Slow tissues have low, shallow-sloped lines close to the diagonal.
Slow tissues have a smaller margin
At the surface, measure the distance between each line and the diagonal. The 5-minute tissue has a margin of about 2.2 bar. The 240-minute tissue has about 0.5 bar.
This margin is the allowed supersaturation, and it decreases as the half-time increases. A tissue that releases gas over hours is allowed much less supersaturation than one that releases it over minutes.
The graph gives the limit for each tissue at any pressure. For planning, a diver needs a limit for each stop depth. Reading Workman’s lines at one depth at a time turns them into a table.
- Ambient pressure
- Margin above it
The limits at the surface
Each bar is one tissue’s limit at the surface. The grey part is the ambient pressure. The coloured part is the margin above it: the excess nitrogen that tissue may hold when you surface.
For the fast tissues most of the bar is coloured. For the slow tissues the coloured part is short.
The limits at a 9 m stop
At a 9 m stop every limit is higher. ΔM is larger for fast tissues, so their limits increase more than the ambient pressure does, and their margin grows.
For the slowest tissues ΔM is only a little above 1. Their limit increases at almost the same rate as the ambient pressure, so their margin hardly changes.
The limits at 30 m
At 30 m the margin of the 5-minute tissue has increased by about 2.4 bar. The margin of the 240-minute tissue has increased by about 0.3 bar.
A single ratio does not account for this: tolerance depends on depth as well as on the tissue. At a deep stop a fast tissue may be allowed much more supersaturation than at a shallow one.
Comparison with Haldane’s ratio
The white ticks show Haldane’s 1.58 ratio at the same 30 m. Workman allows his fast tissues more than Haldane did, and his slow tissues less.
A single ratio cannot do both. For this reason the M-value replaced it.
Try it: find the first stop
Choose one of Workman’s tissues and set how much nitrogen it holds when you leave the bottom. Moving left on the graph is ascending. The diver may ascend until the nitrogen line meets the tissue’s limit, then must stop at the next 10 fsw stop below that point. Compare this with the stop from Haldane’s ratio: shallower for a loaded slow tissue, deeper for a loaded fast one.
- Workman, 20-min tissue
- Haldane 1.58 ratio
- Tissue nitrogen
What to remember
- Workman replaced Haldane’s single ratio with a limit for each tissue: the M-value, the most gas that tissue may hold at a given depth.
- Each limit is a straight line: M0 at the surface, increasing by ΔM for every unit of depth.
- Fast tissues have high, steep lines. Slow tissues have low, shallow-sloped lines close to the ambient pressure, so they are allowed the least supersaturation.
- A stop is the shallowest depth at which every tissue is still below its line.
- Bühlmann, the US Navy’s Thalmann algorithm and gradient factors all keep this straight-line limit.
M-values in the planner
DiveLogic plans with Bühlmann’s M-values, which are derived from Workman’s lines. The tissue view of every plan shows each compartment’s load against its own limit, minute by minute, including the slow tissues that control the last stops.
Sources
- Workman R. D. (1965). Calculation of decompression schedules for nitrogen-oxygen and helium-oxygen dives. Research Report 6-65, US Navy Experimental Diving Unit.
- Boycott A. E., Damant G. C. C., Haldane J. S. (1908). The prevention of compressed-air illness. Journal of Hygiene 8(3): 342-443.
- Baker E. C. (1998). Understanding M-values. Immersed 3(3). (Tabulates Workman’s nitrogen M0 and ΔM values, and restates Haldane’s ratio for nitrogen as 1.58.)
- Gerth W. A. (2010). Thalmann algorithm decompression table generation software design document. NEDU TR 10-09, Navy Experimental Diving Unit. (Workman’s linear expression, still used to project limits to stop depths.)