Rebreather Chemistry

When designing rebreather  the problem of estimation of the mass of  gas flowing through the orifice (nozzle) of the dosing facilities arises. As such unit a pressure reduction valve mostly serves followed by the nozzle through which the gas flows into space where the ambient pressure reigns. Often double nozzles are used to prevent problems with clogging of one of them.  The gas flows obviously from the place of higher pressure p₀  to place of a lower one (p₁) and during this process expands. This process cannot be described by the commonly used equation of state p.v = R.T, which in principle describe a slow sequention of equilibrium states. Now the process is fast, adiabatic, without sharing heat with the surroundings. In the Ideal Gas approximation (it´s also our case) it could be described by the socalled adiabatic equation

p.v^k = R.T.        Here k is also called  Poisson constant given by expression

k =c_p / c_v   which states that k is the ratio of specific heats c measured at constant pressure and constant volume.

The values of k are specific for each gas and from thermodynamics can be by simple reasoning derived  that approximately: 

1,67 for monoatomic molecules  (e.g. He, Ar)

k =       1,40 for biatomic ones ( e.g.O₂, N₂, air, nitrox)

1,33 for tri+atomic (e.g.CO₂ ...)

Experimentally estimated data for gases in our interest can be found in the included table together with their densities (specific masses) r_n at normal conditions  (temperature 0°C, pressure 1bar =10⁵ Pa)  and with their molar masses M_m.

For mixes of gases the relevant values can be obtained by means of the E.g. the value of  k of Nitrox 32/68 can be calculated as k = 0,32 × k_O2 + 0,68 × k_N2 = 0,32 × 1,416 + 0,68 × 1,404 =1,296.

Gas	k	rn(kg/m3)	Mm(kg/kmol)
air	1,406	1,276	28,96
N2	1,404	1,234	28,01
O2	1,416*	1,409	32,00
He	1,630	0,176	4,00
Ar	1,668	1,759	39,95
Ne	1,64	2,277	50,49
CO2	1,304	1,951	44,01
H2	1,41	0,0089

Published values of k lie in the range 1,40 -1,416. The equation of adiabatic change of state can be also expressed in other form: r₀/r = (p₀/p)^1/k =(T₀/T)^1/(k-1) in which r  is density and T is the thermodynamic temperature (in kelvins).

                                                     

Flow velocity:

Consider ideal gas flowing out from a vessel in which its pressure, density and (thermodynamic) temperature are denoted as p₀ ,r₀  and T₀  into the environment of pressure p through an orifice of diameter d as depicted above.

As usually the energy conservation law must be fulfiled expressed in the case of flowing media by the Bernoulli equation. This equation states that in each moment the sum of kinetic and potential energies is constant:

               _p      
w²/2 + ò_p0 (dp/r) +g×h =const.

In this equation the first term represents the kinetic energy (w represent velocity), the second and third ones represent the potential energy. In form of  integral (second term) the pressure energy is expressed, which is of the main importance for gas in contrary to the third term representing the gravitational energy which concerns much more liquids and can be neglected in the case of gas calculations.

Let us suppose that gas starts flowing from a still state ( which is not exactly true as our reasoning concerns the intermediate pressure space into which the gas flows from the high pressure and moreover a little bit cooled by the expansion. The error introduced by the neglection of those facts is allowable). Then the constant  on the right side of the equation can be put equal zero and we can get                            

           _p          
w²/2 + ò_p0 (dp/r) +g×h =const.

Involving the expression for r from the adiabatic equation and solving the integral we obtain for the gas velocity expression

w² = ... = [2.k/(k -1)].p₀/r₀ . [1 -(p/p₀) ^{(k -1)/k}]

This expression is usable to calculate flow at subsonic velocity. The gas velocity in the narrowest („critical“) place of the nozzle cannot be higher than the local sound velocity. But the sound velocity is given by          

c =Ö (dp/dr)

Units

P= pressure

V= volume

R= gasconstant [8,314 J mol^-1 K^-1]

T= temperature in Kelvin

Kelvin = Celsius + 273.15 // Fahrenheit =( Celsius/(5/9)) + 32

k =c_p / c_v= poisson constant

M_m = molar mass

p₀ = pressure before orifice

r₀ = density before orifice

T₀ = temperature in Kelvin before orifice

T_n = normal temperature in Kelvin

T_crit = critical temperature in Kelvin

W = velocity

W_crit =critical velocity

g = acceleration at surface due to Earth's gravity= 9.8 m/s²

h = height above ground

c = characteristic speed

p_crit = critical pressure

p_amb = ambient pressure

r = density

r_krit = critical density

r_n = normal density

B = ratio of the outlet to inlet pressures

S = area section of the narrowest part of the nozzle (S=p × d² /4)

d = orifice diameter

A = constant

B = constant

PRV = pressure reduction valve

CMF = constant mass flow

MCCCR = manual controlled closed circuit rebreather

IP = intermediate pressure

p₀₀ = absolute intermediate pressure at the surface (=p_s+b)

p_0h = intermediate pressure at certain depth (=p_s+p_amb)

D = gas dose in normal (surface) liters per minute

D_h = gas dosis produced by IP at certain depth h

D₀ = gas dosis measured at the surface (open PRV)

D_n = gasflow through orifice