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Mass loss rates and volume leakage rates in laminar or molecular flow regime

Conversion of mass loss rates in pV tracer gas leakage rates is calculated in two steps. First the mass loss rate of the respective medium is converted into a pV leakage rate of the medium. In the second step, medium conversion is calculated according to the formulas valid for the respective gas flow regime. So we can either use formulas for laminar-viscous flow or molecular flow. In case that we do not know which flow regime we are in, we need to calculate both and assume the worse case.

Example:
An air conditioning system must not lose more than 1 g refrigerant R134a per year in order to maintain functionality – what is the helium leakage rate at a temperature of 25 °C?

Mass loss rates and volume leakage rates in laminar flow regime
First step: Conversion of refrigerant mass loss in a refrigerant pV gas flow
  • q_{pV} = q_{M} \cdot \frac{R \cdot T}{M}


qpV
Leakage rate
qM
Mass loss rate
R
Gas constant
T
Temperature (abs)
M
Molar mass
q_{pV, R134a} = \frac{1 \ g/a}{365 \cdot 24 \cdot 3,600 \ s/a} \cdot \frac{83.14 \ mbar \cdot l \cdot mol^{-1} \cdot K^{-1} \cdot 298 \ K}{102 \ g \cdot mol^{-1}} = 7.7 \cdot 10^{-6} \ mbar \ l / s

q_{pV, R134a} = 7.7 \cdot 10^{-7} \ Pa \cdot m^{3} \cdot s^{-1}
Second step: Conversion of refrigerant pV gas flow into a helium pV gas flow (i.e. leakage rate)
  • \frac{q_{\mathrm{Gas1}}}{q_{\mathrm{Gas2}}} = \frac{\eta_{\mathrm{Gas2}}}{\eta_{\mathrm{Gas1}}}

q_{pV, \ Helium} = q_{pV, \ R134a} \cdot \frac{\eta_{R134a}}{\eta_{Helium}} = 7.7 \cdot 10^{-7} \ Pa \cdot m^{3} \cdot s^{-1} \cdot \frac{11.6 \ \mu Pa \ s}{19.1 \ \mu Pa \ s} = 4.7 \cdot 10^{-7} \ Pa \cdot m^{3} \cdot s^{-1}


Mass and volume leakage rates in molecular flow regime
In the example above we have assumed laminar flow regime. So what happens in molecular flow regime?

Example:
An air conditioning system loses 0.01 g refrigerant R134a per year – what is the helium leakage rate under identical pressure conditions at a temperature of 25 °C?


q_{pV, \ Helium} = q_{pV, \ R134a} \cdot \sqrt{\frac{M_{R134a}}{M_{Helium}}}
= q_{M, \ R134a} \cdot \frac{R \cdot T}{M_{R134a}} \cdot \sqrt{\frac{M_{R134a}}{M_{Helium}}}


q_{pV, \ Helium} = \frac{0.01 \ g/a}{3.15 \cdot 10^{7} \ s/a} \cdot \frac{83.14 \ mbar \cdot l \cdot mol^{-1} \cdot 298 \ K}{102 \ g \cdot mol^{-1} \cdot K} \cdot \sqrt{\frac{102}{4}} = 3.9 \cdot 10^{-7} \ mbar \ l/s


q_{pV, \ Helium} = 3.9 \cdot 10^{-8} \ Pa \cdot m^{3} \cdot s^{-1}