Lumped-Parameter Response Time Models for Pneumatic Circuit Dynamics

Resistor–capacitor (RC) response time models for pressurizing and depressurizing a pneumatic capacitor (mass accumulator) through a resistor (flow restriction) comprise a framework to systematically analyze complex fluidic circuits. A model for pneumatic resistance is derived from a combination of fundamental fluid mechanics and experimental results. Models describing compressible fluid capacitance are derived from thermodynamic first principles and validated experimentally. The models are combined to derive the ordinary differential equations that describe the RC dynamics. These equations are solved analytically for rigid capacitors and numerically for deformable capacitors to generate pressure response curves as a function of time. The dynamic pressurization and depressurization response times to reach 63.2% (or $1−e−1$⁠) of exponential decay are validated in simple pneumatic circuits with combinations of flow restrictions ranging from 100 μm to 1 mm in diameter, source pressures ranging from 5 to 200 kPa, and capacitor volumes of 0.5 to 16 mL. Our RC models predict the response times, which range from a few milliseconds to multiple seconds depending on the combination, with a coefficient of determination of $r2=0.983$⁠. The utility of the models is demonstrated in a multicomponent fluidic circuit to find the optimal diameter of tubing between a three-way electromechanical valve and a pneumatic capacitor to minimize the response time for the changing pressure in the capacitor. These lumped-parameter models represent foundational blocks upon which timing models of pneumatic circuits can be built for a variety of applications from soft robotics and industrial automation to high-speed microfluidics.