Gas flow through micro-capillaries - which flow law is most suitable to predict the flow rate through micro-capillaries?

In pharmaceutical quality control, capillaries are the primary positive controls able to cover a wide range of gas flow rates during Container Closure Integrity (CCI) testing. They serve as the physical benchmarks for analytical methods to demonstrate these can reliably detect defects at or below the critical leak thresholds. However, the predictive reliability of standard gas flow models for capillaries can be undermined by the 'transitional' flow regime. Traditional flow laws, like Hagen-Poiseuille and molecular flow, provide accuracy only at the extremes of the investigated flow rate spectrum, failing to deliver precise results in the range most relevant for the sub-micron defects that define current pharmaceutical safety standards. This study investigates helium flow through capillaries with inner diameters from sub-micron to 25 μm corresponding to the range of leak rates from 10-9 to 10-3 mbar l/s, encompassing the 6·10-6 mbar l/s mentioned in USP<1207>. We evaluated the predictive performance of classical, slip-flow, and modified Knudsen models under differential pressure ranging from 250 mbar to 4000 mbar, specifically comparing behavior at a constant 1000 mbar difference against variable pressure dependencies. Our analysis reveals that model strength is highly context-dependent. At a constant 1 bar pressure difference (ambient vs. vacuum), the slip-flow model was the superior predictor, maintaining a 30% tolerance and outperforming the other flow laws including the modified Knudsen law model by Yoshida et al.. However, for characterizing pressure dependence, simpler models failed to capture the transition from linear ~p in ø0.9 μm to quadratic ~p2 in ø 20 μm capillaries. In this aspect the modified Knudsen model excelled, accurately describing the shift in behavior across the entire diameter range. These results provide a robust framework for CCI testing, enabling industry to select models to simulate specific relevant leak limits under varying environmental and pressure-test conditions.