# Volumetric & Mass Flow Rate Through a Piston-Cylinder Annular Gap with Laminar Flow?

I am attempting to define the proper equations used to calculate both volumetric and mass flow (based on pressure differential) through a Laminar Flow Element. The element consists of a piston concentrically mounted within a cylindrical bore, resulting in an annular flow path. I have found that using Poiseuille's Law may be limited due to the difficulty of defining the hydraulic radius required for that equation.

I have found an equation that may be applicable from a MIT white paper, but my usage of dP in this equation results in very small volumetric flow rates (See equation below).

$$Q = \dfrac{A\cdot \text{d}P\cdot D^2}{32L\mu}$$

where

• $$D$$ = Annular Gap between piston and bore
• $$A$$ = Cross-sectional area of flow path
• $$\text{d}P$$ = Pressure drop across element
• $$L$$ = Length of restriction
• $$\mu$$ = Dynamic (absolute) viscosity of gas at flow conditions

Is this the correct formula to use in calculating volumetric flow rate based on $$\text{d}P$$ through the element?

If considering the following parameters for the laminar flow element and the gas tested is air, what should the volumetric flow rate be? Also, what would the resulting mass flow rate and Reynold's Number be?

• Piston OD: 0.003175 m
• Cylinder Bore ID: 0.00345 m
• Annular Gap: 1.375E-4 m
• Characteristic Length: 0.00794 m
• $$\text{d}P$$ across annular restriction: 3.5 "H2O = 871 Pa**

** - This is where I am having issues. How does this value correlate to the atmospheric conditions of one end of the element? In using a posteriori, I calculate a volumetric flow rate of 3.0E-6 m^3/s ~= 181 cc/min, which is not the set flow of the source.