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# Hydrogen-HeliumSimilarity

### Buoyancy effect in the mean flow

To investigate on the relation between helium and hydrogen buoyancy we need to consider the non-dimensional form of the buoyancy term in the vertical momentum equation. If we introduce L and U as the reference length and velocity scales, then this term takes the form of a local Richardson number, expressing the ratio between buoyancy and inertial forces: $$Ri_{local}={{(\rho_a-\rho) g L} \over {\rho U^2}}$$ In this analysis we shall use the Boussinesq approximation (mass fraction f<<1), which implies replacing the mixture density with the density of ambient air in all the terms of the mean flow conservation equations except for the density difference in the buoyancy term. Additionally by introducing the local molar concentration C the above buoyancy term becomes proportional to a global Richardson number Ri : $$Ri_{local}= C Ri$$ where $$Ri= {{(\rho_a-\rho_g) g L} \over {\rho_a U^2}}$$

The relation between helium and hydrogen buoyancy can now be assessed using this global Ri, assuming same velocity and length scales: $${{Ri_{He}}\over{Ri_{H2}}}= {{\omega_{He}-1} \over {\omega_{H2}-1}} {{\omega_{H2}} \over {\omega_{He}}} \approx 0.92$$

The proximity of the ratio of the Richardson numbers to 1.0 justifies the use of He instead of H2 for molar concentration measurements in regions of the flow were the Boussinesq approximation is valid. This result can be compared to Agranat et al., 2004 Invalid BibTex Entry!, who derived the ratio of the Richardson numbers to be 0.47 (compared to 0.92 here), which is strongly different to the present result and certainly does not justify performing hydrogen dispersion tests using helium. The reason for this discrepancy is that Agranat et al. defined their Richardson number by dividing with the density of the gas and not of the air as has been done in the present analysis. Use of the density of the gas could be justified only where the mixture is very rich, e.g. very close to the source. In these regions of course the Boussinesq approximation is not valid.

### Buoyancy effect in the turbulent kinetic energy

Buoyancy affects the mean flow directly and indirectly. The direct influence is through the buoyancy term in the vertical momentum equation as shown in the above paragraph. The indirect influence is through the buoyant production term of the turbulent kinetic energy (TKE) equation. With the eddy viscosity approach and the introduction of the molar concentration this term is: $$G=-\rho w g = {\nu_t \over Sc_t} {{\partial \rho} \over {\partial z}} g = - {\nu_t \over Sc_t} {{\partial C} \over {\partial z}} g (\rho_a - \rho_g)$$

By introducing, L and U as reference length and velocity scales and under the Boussinesq approximation the non-dimensional form of the buoyant production term becomes proportional to the Richardson number and inversely proportional to the turbulent Schmidt number: $${{G} \over {\rho_a U^3 / L}} = - {\nu_t \over {U L} } {{\partial C} \over {\partial z}} {Ri \over Sc_t}$$

This result shows that the effect of the released substance on the buoyant production term can be investigated using the ratios of Richardson and turbulent Schmidt numbers. The Richardson ratios were close to 1.0 as shown in the above paragraph. The turbulent Schmidt number is usually taken as 0.7 and independent of the substance. Therefore, under the Boussinsq approximation the effect on the buoyant TKE production term of replacing hydrogen with helium in hydrogen dispersion tests is considered to be small. This is not valid for regions of the flow were the Boussinesq approximation does not hold.

### Axial concentration in axi-symmetric jets

Using the definitions and correlations introduced above for axi-symmetric vertical jets we can compare the axial molar concentrations for helium versus hydrogen releases. It can be easily shown that the helium to hydrogen Froude ratios are: $${{F_{He}}\over{F_{H2}}} = {{\omega_{H2}-1}\over{\omega_{He}-1}} \approx 2.2$$ Then, one can obtain the helium to hydrogen mass fraction ratios as follows: In the non-buoyant jet region: $$\left( {{f_{He}}\over{f_{H2}}} \right)_{NBJ} = \left( {{\omega_{H2}}\over{\omega_{He}}} \right)^{1 \over 2} \approx 1.4$$ In the intermediate buoyant jet region: $$\left( {{f_{He}}\over{f_{H2}}} \right)_{BJ} = \left( {{\omega_{H2}-1}\over{\omega_{He}-1}} \right)^{1 \over 8} \left( {{\omega_{H2}}\over{\omega_{He}}} \right)^{1 \over 2} \approx 1.6$$ In the buoyant plume region: $$\left( {{f_{He}}\over{f_{H2}}} \right)_{BP} = \left( {{\omega_{H2}-1}\over{\omega_{He}-1}} \right)^{1 \over 3} \left( {{\omega_{H2}}\over{\omega_{He}}} \right)^{2 \over 3} \approx 2.1$$

Under the Boussinesq approximation, the helium to hydrogen molar fraction ratios become: $$\left( {{C_{He}}\over {C_{H2}}} \right) = \left( {{f_{He}} \over {f_{H2}}} \right) \left( {{\omega_{H2}}\over{\omega_{He}}} \right)$$

Then for the non buoyant jet region: $$\left( {{C_{He}}\over {C_{H2}}} \right)_{NBJ} \approx 0.71$$ In the intermediate buoyant jet region: $$\left( {{C_{He}}\over {C_{H2}}} \right)_{BJ} \approx 0.81$$ In the buoyant plume region: $$\left( {{C_{He}}\over {C_{H2}}} \right)_{BP} \approx 1.03$$ The higher the proximity of these ratios to 1.0 to more justified is the use of He instead of H2 for molar concentration measurements.