The following text is written under the assumption that the reader has thorough knowledge about the relation between gas fractions and partial pressures as well as basic knowledge in rebreather design. Also, Nitrox concepts like MOD and EAD are assumed to be familiar to the reader. A few differential equations and Laplace transforms appear but it is not necessary to understand them in order to understand constant mass flow semi-closed rebreather (CMF SCR).
The text presents the general behaviour of the breathing gases in a CMF SCR as well as two ways of determining the fresh gas flow and fresh gas mixture in a CMF SCR, one based on a known fresh gas mixture and one based on a planned diving depth.
Note that different authors and different manufacturers in general use their own values for the maximum and minimum oxygen uptakes as well as the safety factor when dimensioning the flows. It is up to the reader to use ****his/hers*** knowledge and common sense to determine whether he/she believes that the estimates used are relevant or not!
/*
begin US lawyer BS */
Please note: The following pages most likely contains a lot of errors and
misunderstanding that could mislead you and make you build and or use dangerous
devices that might get you, or someone you like, killed. If you believe in any
of the information found on these pages it is your own fault! The author
assumes no responsibility whatsoever neither for errors in the text nor for
possible mishaps because a reader misreads or misinterprets the information
contained herein.
/* end US Lawyer BS
*/
Contents
Constant Mass Flow
Semi-closed Rebreather theory
Calculation of
oxygen uptake during a dive
Maximum Operational
Depth, MOD
MOD calculated from the fresh gas
mixture in the tank
MOD calculated from gas concentration in the loop with the diver at
rest.
Dimensioning of
flows based on known fresh gas fraction
The Flow and the safety factor
The oxygen uptake (or oxygen consumption), _{}, of a diver vary with how hard the diver is working.
Oxygen uptake |
Equivalent work |
0.3 [L/min] |
The oxygen uptake at rest for an average 75 kg man |
0,8 [L/min] |
The oxygen uptake when swimming at about 0.5 knots. |
1.3 [L/min] |
The average long time consumption of a fit hard working diver. |
2.0 [L/min] |
The maximum oxygen uptake used by a diver* |
* Note that a very fit diver can, in
extreme cases, consume more than 4 [L/min] swimming in water but oxygen
consumptions as high as 2 [L/min] are rarely seen in real dives.
The reader that really is interested in the relevant physiology is referred to: Bennet and Elliot, The Physiology and Medicine of Diving, Saunders 1993, ISBN 0-7020-1589-X, especially chapter 5 respiration and exertion.
In a semi closed rebreather the oxygen fraction in the breathing circuit, F_{O2}, depends on the diver's oxygen uptake, V_{O2}, and the added flow of fresh gas, Q_{mix}, and fresh gas oxygen fraction (i.e. percentage), F_{mix}, of the fresh gas.
Figure
1. A simplified model of a
semi-closed rebreather breathing system.
In the Figure 1 above the following parameters are used:
The change of the amount of O_{2} in the system during the time dt can be expressed as:
_{} Eq. A
The differential equation describing the system then becomes:
_{} Eq. B
This differential equation is solved using Laplace transformation:
_{}
_{} Eq. C
Which transfers back into the time domain and has the solution:
_{} Eq. D
The equation comprises of a steady state term to the left and a transient tem to the right. The steady state term to the left is frequently found in SCR RB literature:
_{} Eq. E
The nominator describes the net flow of oxygen flow in the circuit and the denominator the total flow leaving (i.e. venting) the breathing circuit.
The time constant, i.e. how fast the oxygen fraction change, of the semi-closed rebreather depends on the flow venting the breathing circuit and the gas volume (as measured in normal litres, i.e. surface equivalent volume!) of the complete breathing circuit, i.e. including divers lungs, canister, tubes, and counter lungs.. The time constant can be seen as the inverse of the exponent in the right hand part of the Eq. D.
_{} Eq. F
The time constant, t, i.e. the time it takes for the system to change, is significant, see the figure below. The time constant is the time needed for a system that sees a rapid (i.e. a step) change to reach 63% if the final (i.e. steady state) value. About 3 time constants is needed for the system to reach within 1% of the steady state value.
Figure 2. The time constants for a constant mass flow semi-closed rebreather. The fresh gas flows are from the Draeger Dolphin manual for the respective fresh gases. The oxygen consumption is assumed to be 1.3 [L/min] and the total loop volume 12 liters.
Figure 3. An illustration of the rate of change in oxygen fraction in a
breathing circuit at different depths. The starting fraction is 40%, the total
volume of the circuit is 12L, the oxygen uptake of the diver is 1.3 L/min, the
fresh gas mixture is 40%, and the fresh gas flow is 10.3 L/min. The steady
state oxygen fraction in the breathing circuit is 31.4%.
The deeper the CMF SCR rebreather is dived, the
slower the oxygen fraction changes.
In practise the result is that a semi-closed rebreather that is dived from the surface straight down to 50m and staying there for 10 min exposure time will not have reached the steady state oxygen fraction value when leaving the bottom!
For a SCR with a fresh gas flow of 12 [L/min], a fresh gas oxygen fraction of 32.5%, and an oxygen uptake of 1,3 [L/min], the fraction in the circuit will be about 25% when leaving the bottom, see simulation in the graph below.
Figure 4. Simulation of a 10 min dive to 50 m with a CMF SCR running 32.5% at 12 L/min. The oxygen uptake is 1.3L/min. Ascent and descent rates 10m/min. Note the dropping in F_{O2} suring ascent due to the increased loss (i.e. dumping) of oxygen.
Note:
The rate of change in oxygen fraction in a Closed Circuit Rebreather, CCR, does
_NOT_ behave like the CMF SCR!
The Equation E above can be used to estimate the oxygen uptake. The equation is then solved for V_{O2}:
_{}
Eq. G
Or rewritten using depth, D, [in meters] and measured oxygen partial
pressure, P_{O2}^{meas}
_{}
where the (D+10)/10 is the absolute pressure
[bar] at a certain depth, D, [in meters]
_{} Eq. H
By measuring the oxygen partial pressure in the breathing loop, P_{O2}^{meas} and calculating the oxygen fraction during the dive the oxygen uptake can be estimated. _BUT_ there is a great risk of making a significant error here. If, for example, the diver descends from the surface down to 10m and rests there while trying to estimate his oxygen consumption his calculations will result in a underestimation of his uptake since the rebreather has not yet reached steady state conditions after the descent and bypass, see also the paragraph on time constants above.
If this erroneously low estimate of oxygen consumption is used as a rationale for lowering the fresh gas flow in a CMF SCR there is a true risk for a hypoxic surprise when working hard.
I strongly advice against using the oxygen consumption calculated from readings during a dive as a rationale for lowering the constant mass flow in your rebreather!
There are (at least) two practices taught when
calculating the MOD for a CMF SCR. The first one uses the oxygen fraction of
the tank mixture like any OC Nitrox and the second calculates the oxygen
fraction in the loop with the diver at rest.
It is up
to the reader to choose which way he/she prefers. In practice the difference in
MOD calculated using the two methods is very small.
The maximum diving depth is determined by the
fact that the oxygen partial pressure in the breathing gas should not exceed a
certain limit. In cold waters a max P_{O2} of 1.4 [bars] is often used.
_{} Eq O.
_{} Eq. P
An example with an F_{mix} of 40% and P_{O2}^{max} of 1.4 bar:
_{}
The highest oxygen fraction is seen
in the loop when the diver is at rest with low oxygen consumption; a V_{O2}^{min}
of 0.25 [L/min] is often used.
_{} [bar] Eq. R
Or recalculated into depth in
meters:
_{} [m] Eq.
S
Continuing the example using Eq. R:
_{}
The oxygen fraction in the loop at
rest will be about 38% but note that the oxygen fraction in the loop immediately after a descent will
be higher than the steady state value partly because of the gas added
for volume compensation using the bypass.
100
Note that different authors and different manufacturers use different estimated values for the maximum and minimum oxygen uptakes as well as different safety factors when dimensioning the flows. It is up to the reader to use his knowledge and common sense to determine whether he/she believes that the estimates and assumptions used below are relevant or not!
The necessary fresh gas flow, Q’_{mix}, is calculated so that the resulting (minimum) oxygen fraction, F_{O2}, in the breathing circuit will be 0.2 (i.e. 20%) when the divers’ oxygen uptake, V_{O2}, is 2 [L/min].
The fresh gas flow can be expressed in known variables as:
_{}
_{} Eq. I
In order to determine the fresh gas flow, Q_{mix}, to use in the rebreather this value for the necessary Q’_{mix} should be increased with a safety factor, k, which usually is in the range of 0.25 to 0.5 (i.e. 25% – 50%).
_{}
An example using an F_{mix} of 40% and a safety factor, k, of 0.25:
_{}
_{}
Using a 25% safety factor for a 40% mix the results indicate that you should use 10 [L/min] of fresh gas flow, see table 1 below.
F_{mix} [%] |
Q_{mix} (k=0) [L/min] |
Q_{mix}
(k=0.25) [L/min] |
Q_{mix}
(k=0.5) [L/min] |
32,5% |
12,8 |
16,0 |
19,2 |
40,0% |
8,0 |
10,0 |
12,0 |
50,0% |
5,3 |
6,7 |
8,0 |
60,0% |
4,0 |
5,0 |
6,0 |
80,0% |
2,7 |
3,3 |
4,0 |
Table 1. Exemplifying the flow
calculated from known fresh gas oxygen concentration and with different safety
factors.
The relation between partial pressure, P_{XX}, gas fraction, F_{XX}, and total pressure P_{tot}:
_{}
Conversion of depth (in meters) to the corresponding pressure in bars:
_{} [bar]
Conversion of pressure in bars to the corresponding depth in meters:
_{}
[m]
Calculation of the Equivalent Air Depth (EAD) from known Nitrox gas mixture and depth.:
_{} [m]
Where F_{N2} is the fraction of Nitrogen in the gas.
Calculation of depth from a known Nitrox mixture and desired Equivalent Air Depth (EAD):
_{} [m]
Calculation of the maximum operational depth for a gas mixture based on maximally acceptable oxygen partial pressure, P_{O2}^{max}.
_{} [m]
All calculations in this document are metric but the fan of imperial units can easily find conversion tools on the Internet.
In general, a capital letter is used to denote a physical variable; an index is used to specify the variable and an exponent as well in some specific case.
P |
Pressure |
F |
Fraction |
V |
Volume |
_{} |
(Volume flow but the dot is tricky to write in word outside the equation editor. |
Q |
Flow (mass flow) |
T |
Time |
D |
Depth |
Some examples of indexes used in this text:
O_{2} |
Oxygen |
N_{2} |
Nitrogen |
mix |
Mixture, usually referring to the fresh gas of the gas delivery. |
tot |
Total |
max |
Maximum |
min |
Minimum |
avg |
Average |
meas |
Measured |
Last updated 2004-09-30
All material on this website is copyright 2002-2004 by Åke Larsson. All rights reserved.