Principles of Constant Mass Flow
or
How does the Dräger Dolphin Work?
Nevil Adkins

Note: the following information is mainly theoretical in
nature, has not been checked by others, is not issued by Dräger and may not
be 100% correct. As an engineer I have applied my knowledge of
thermodynamics and hopefully this will give an insight as to how the Dolphin
works.
Introduction
The Dräger Dolphin and its stable mate the Ray are
recreational Semiclosed Circuit Rebreathers (SCRs) using a constant
massflow design to add gas to the breathing loop. They are very similar in
nature so I will talk of the Dolphin and highlight where the Ray is
different where I can (I use the Dolphin, not the Ray so may not know all
the differences). The basic nature of the beast is taught on the relevant
user course and will not be dwelt on here. For further reference read the
user manuals supplied with the units and “Rebreather Diving”[1]
by Bob Cole.
Many questions have been raised on various email groups
about how the Dolphin will react when changing the interstage pressure,
diving deep, using different gas mixes, etc. The explanations that have
ensued have been vague, confusing or even wrong in some circumstances so I
will attempt to explain the engineering theory that drives the Dolphin. The
theory is quite simple in principle but to really get to terms with it you
need some heavyduty maths. For those who want the inside story refer to
“Engineering Thermodynamics”[2]
by Rogers and Mayhew. The relevant chapter is Chapter 18 – One Dimensional
Steady Flow and Jet Propulsion.
Background theory
Let us take a basic look at the theory of gas flow in nozzles. Consider the
situation in figure 1


Gas in a large container has an initial pressure p_{1},
temperature T_{1} and velocity v_{1} which is negligible.
The ambient pressure at the outlet of the nozzle is p_{2}. Since p_{2}
is lower than p_{1}, the gas feels a pressure difference resulting
in an accelerating force. The gas begins to move towards and through the
nozzle. The gas’s potential energy in the form of pressure is gradually
converted to kinetic energy in the form of velocity. As the inlet pressure
is increased, the gas feels a greater acceleration force and the velocity
achieved in the throat of the nozzle increases. At a certain point however
the difference in pressure between p_{1} and p_{2} is
sufficient to accelerate the gas to the speed of sound at local conditions
in the throat of the nozzle (p_{t}, T_{t}), a barrier that
cannot be passed. A further small increase in the initial pressure p_{1}
(potential energy) cannot increase the velocity (kinetic energy) in the
throat so the gas still has some potential energy left – the throat pressure
p_{t} rises above the ambient pressure p_{2}. As the gas
exits the nozzle it becomes unconstrained and expands irreversibly to the
ambient pressure p_{2}, decelerating instantly to low velocity. If
at this stage the ambient pressure were to be lowered then the gas
conditions in the nozzle do not change and no extra flow occurs. Similarly
if the ambient pressure is raised, as long as it does not rise to match the
throat pressure again, no influence on the throat conditions are felt and
the flow rate remains constant. This situation is known as choked flow and
achieves constant mass flow for given inlet conditions, regardless of how
the outlet conditions vary.
What happens if the inlet pressure is increased further?
Since only a fraction of the initial potential (pressure) energy is needed
to accelerate the gas to the speed of sound (kinetic energy), more potential
energy is left over and the pressure in the throat rises. This increases
both the local speed of sound and the gas density, which results in a
greater mass flow. Thus, once choked flow is achieved, variations in outlet
(ambient) pressure will have no effect on the mass flow rate, but variations
in inlet pressure will cause corresponding variations in the mass flow rate.
The rigorous mathematical analysis in Engineering
Thermodynamics leads to the equation

Equation 1

From this equation it can be seen that the critical mass
flow for a particular nozzle (i.e. fixed A_{t}) is a function of the
inlet pressure and temperature and the nature of the gas passing through it.
Critical mass flow is achieved provided that

equation 2

where p_{2} is the nozzle outlet pressure.
From this equation it can be seen that the critical
pressure ratio (p_{2}/p_{1}) is a function solely of the
nature of the gas passing through the nozzle.
What happens if the critical pressure ratio is
exceeded?
Provided that the pressure ratio p_{2}/p_{1
}is less than the critical value then the nozzle will be choked and a
constant mass flow regime will be achieved. What happens if the critical
ratio is exceeded? The answer is that the velocity in the throat of the
nozzle will not reach local sonic and the nozzle will not be choked. The
flow rate will gradually reduce until no flow occurs when the pressure ratio
is unity, i.e. there is no pressure difference. The characteristic is
illustrated in the graph, figure 2.


What is γ?
The physical property of the gas known as γ, is the ratio
of the specific heat capacities at constant pressure and volume and can be
measured for each gas. The γ for a mixture of gases such as nitrox can be
calculated if the composition of the gas is known and the γ for each
component is known.
For
example:

Table 1
Oxygen 

γ = 1.416 
Nitrogen 

γ = 1.403 
Air 
(0.21
x 1.146)+(0.79 x 1.403) 
γ =
1.406 
EAN32 
(0.32
x 1.146)+(0.68 x 1.403) 
γ =
1.407 
EAN40 
(0.4
x 1.146)+(0.6 x 1.403) 
γ =
1.408 
EAN50 
(0.5
x 1.146)+(0.5 x 1.403) 
γ =
1.410 
EAN60 
(0.6
x 1.146)+(0.4 x 1.403) 
γ =
1.411 

How does this theory relate to the Dolphin?
The gas contained in the cylinder of the Dolphin is
reduced to a fixed interstage pressure by the first stage regulator. The
output of this first stage is held constant at 250psi/17.2bar gauge,
irrespective of depth. This contrasts to the usual open circuit and Ray
regulator design, which maintains the interstage pressure at a fixed margin
(typically 10bar/140psi) above the ambient pressure. The gas is then taken
to the dosage jets which each contain two laser drilled rubies that act as
nozzles according to the above theory. The holes in the rubies are sized to
give a certain flow rate with a certain gas mix, within a certain tolerance
band.
From
the basic theory outlined above let us take Equation 1


and deduce the consequences.
1.
T_{1} is the absolute inlet temperature to the jets, measured
in Kelvin. Due to expansion through the regulator it is likely to be a few
degrees cooler than ambient, but probably not significantly. For the most
part we dive in water temperatures between say 273 K (0°C) and 303 K (30°C),
a variation of ±5% from the average 288 K (15°C). This variation will only
change the mass flow by ±2.5% so can probably be ignored totally, and
certainly over the duration of a single dive.
2.
The mass flow is related in a complex way to the properties of the
mix flowing through the nozzle. Whilst considering nitrox, the variation in
γ between EAN32 and EAN60 is about 0.2%, which leads to a variation in the
mass flow of about 0.08%. Even dramatically changing nitrox mixes will not
affect the mass flow rate flowing through any particular jet.
3.
The mass flow is directly proportional to the inlet (interstage)
pressure, p_{1}. Any variation in the interstage pressure, whether
by intent or mishap, will directly affect the mass flow rate. For example,
dropping the interstage pressure from 18.2 bar absolute to 13.6 bar absolute
will reduce the mass flow rate of each of the jets by 25%.
Now let
us take Equation 2


and
deduce the consequences.
1.
If the interstage pressure p_{1} is fixed at 18.2 bar
absolute, then the mass flow will remain constant provided the ambient
pressure p_{2} is less than a certain value, dependant on γ. For
nitrox mixes p_{2}/p_{1} must be less than 0.526. This
implies that the maximum ambient pressure for critical mass flow is 9.6bar
absolute, or approximately 86m. This is way in excess of the 40m limit for
recreational use or 52m limit for military use recommended by Dräger.
2.
If the interstage pressure p_{1} is allowed to vary, either
by accident or design, the maximum depth achievable whilst retaining
critical flow will also vary.
How does this theory relate to the Ray?
Unlike the Dolphin, the first stage of the Ray reduces
the pressure in the cylinder to an interstage pressure at a fixed margin
above ambient pressure – this is the same as most open circuit regulators.
The gas is then taken to the dosage jet, there is only one, which like the
Dolphin contains two laser drilled rubies that act as nozzles according to
the above theory. The Ray is specified for diving to a maximum of 22m with
EAN50.
From
the basic theory outlined above let us take Equation 1


and deduce the consequences.
1.
As for the Dolphin, the variation in flow rate with temperature is
miniscule and can effectively be ignored.
2.
As with the Dolphin, the mass flow is related in a complex way to the
properties of the mix flowing through the nozzle. Since the Ray is
specified for use only with a single mix there should be no variation in γ
but as we have seen for the Dolphin the variations in γ, and the subsequent
variations in the flow rates are also negligible with varying nitrox mixes.
The Ray could therefore be used with a number of different mixes, albeit
with the limitation of choice of flow rates.
3.
The mass flow is still directly proportional to the inlet (interstage)
pressure, p_{1}, but unlike the Dolphin this is not fixed. The jet
inlet pressure at is related to the depth in the following manner:


equation 3

which for a typical interstage pressure of 10bar becomes

equation 4

Referenced to the flow rate at the surface, the flow rate
at the specified maximum depth of 22m will be 20% higher. The gas
consumption of the Ray is therefore not independent of depth as the Dolphin
is but the reduction in unit duration from 1½ hours at the surface to 1¼
hours at 20m is insignificant for the majority of recreational users at whom
this unit is aimed.
Now let
us take Equation 2


which
for nitrox reduces to

equation 5

Solving
Equations 3 & 5 gives


which
for our typical interstage pressure of 10 bar means P_{ambient}
≤ 11.1bar or 101m

The Ray will not stray out of the critical ratio
limitation until extreme depth – but don’t forget the mass flow is not
constant across the depth range! The Ray is still not suitable for diving
much beyond the recommended maximum of 22m.
Does the Dolphin/Ray use “supersonic” flow?
No. The gas passing through the nozzle accelerates from
negligible velocity until it achieves sonic velocity at local conditions of
temperature and pressure. Theoretically the gas could be further
accelerated as it leaves the nozzle, to achieve supersonic velocities, but
this would not increase the mass flow. Furthermore the nozzle would have to
be very carefully (and expensively) engineered, and very minor variations in
conditions would quickly bring the nozzle off its design point. The gasses
leaving the nozzle expand instantaneously and irreversibly, and decelerate
back to negligible velocity. Supersonic velocities are not needed.
How will using trimix affect the Dolphin?
Modern thinking is that diving beyond 40m should be done
using trimix. The section on the Dolphin above shows that the maximum
theoretical depth for the Dolphin should be in the range of 80m. How will
using trimix in the Dolphin in the 4080m range affect performance?
Returning to equation 2 reminds us that the critical pressure ratio is
dependant on γ, which is itself a function of gas composition. Extending
table 1 to include helium and trimix gives

Helium 

γ = 1.630 
TMX27/30 
(0.27
x 1.146)+(0.43 x 1.403)+(0.3 x 1.630) 
γ =
1.475 
TMX16/45 
(0.16
x 1.146)+(0.39 x 1.403)+(0.45 x 1.630) 
γ =
1.508 
TMX10/80 
(0.10
x 1.146)+(0.1 x 1.403)+(0.8 x 1.630) 
γ =
1.586 

table 2

It can be seen that
even very high helium mixes such as TMX10/80 only have a γ about 12% higher
than EAN32. The increase in γ reduces the critical pressure ratio slightly,
making the maximum ambient pressure for critical mass flow 9.1bar absolute,
or approximately 81m – not greatly different from the 86m theoretical limit
calculated for nitrox. The difference is that 81m is a feasible dive on
trimix but not for nitrox!
The increase in γ will also be reflected by an increase
in mass flow rate of about 4% for any particular jet. This is well within
the tolerance band of individual dosage jets and below the measurement
accuracy of most people and will not realistically affect dive planning.
This theory is confirmed by several people’s experience in using the unit on
trimix.
The conclusions from these calculations match up
interestingly with the reports that Dräger have developed a modification
pack to allow the Dolphin to be used to 80m on trimix.
How would feeding the dosage jets from a standard
regulator affect the Dolphin?
It has been suggested to feed gas to the dosage jets from
a second source via a standard regulator. In this way it would be possible
to feed a rich nitrox or pure oxygen to the loop during decompression. As
noted in the section regarding the Ray, the use of depth compensating
regulators will result in variable flow rates. Since decompression using
rich mixes is usually only carried out over a small range of depths, the
variation would not be that great and could be coped with. If however the
gas fed to the jets was a bottom gas, say from a sideslung cylinder acting
as either a bailout or as a main cylinder, then the system would be very
dynamic in nature. If set up to give correct flow rates at the surface,
large quantities of gas would be wasted at depth. If set up to give correct
flow rates at depth then the oxygen mass flow rate at shallow depths might
not be sufficient to meet demands with dire consequences. As per the Ray,
use of depthcompensating regulators to feed gasses to the dosage jets
should not be considered deeper than about 22m.
I hope
this look at the physics driving the Dolphin and Ray has been readable, and
more importantly correct. If anybody wishes to offer corrections,
suggestions for further areas of coverage, etc. then please contact me at
adkinsoman@yahoo.com

[1] Rebreather
Diving, Bob Cole, published by SubAqua Association 1998, ISBN 0 9519337
9 5.
[2] Engineering
Thermodynamics – Work & Heat Transfer 4^{th} Edition, Rogers &
Mayhew, published by Longman Scientific & Technical, ISBN 0582045666.

