Relaxation Effects in Small Critical Nozzles

We computed the flow of four gases (He, $N2$⁠, $CO2$⁠, and $SF6$⁠) through a critical flow venturi (CFV) by augmenting traditional computational fluid dynamics (CFD) with a rate equation that accounts for $τrelax$⁠, a species-dependent relaxation time that characterizes the equilibration of the vibrational degrees of freedom with the translational and rotational degrees of freedom. Conventional CFD $(τrelax=0)$ underpredicts the flow through small CFVs (throat diameter $d=0.593mm$⁠) by up to 2.3% for $CO2$ and by up to 1.2% for $SF6$⁠. When we used values of $τrelax$ from the acoustics literature, the augmented CFD underpredicted the flow for $SF6$ by only 0.3%, in the worst case. The augmented predictions for $CO2$ were within the scatter of previously published experimental data $(±0.1%)$⁠. As expected, both conventional and augmented CFD agree with experiments for He and $N2$⁠. Thus, augmented CFD enables one to calibrate a small CFV with one gas (e.g., $N2$⁠) and to use these results as a flow standard with other gases (e.g., $CO2$⁠) for which reliable values of $τrelax$ and the relaxing heat capacity are available.