Volumetric Flow Rate

Another quantity of interest in pipe flows is the volumetric flow rate, which is obtained by integrating the velocity profile. Considering a disc of thickness dr at a radius r we have

$$dQ = (2 \pi r dr) u$$

$$Q = \int_0^R 2 \pi V_c \left [ 1 - \left(r \over R \right )^2 \right] r dr$$

$$Q = { {\pi R^2 V_c}\over 2}$$ (7.13)

If we now define an average velocity, V, such that $Q = \pi r^2 V$ we can verify that

$$V = { V_c \over 2} = {{\Delta p D^2} \over {32 \mu l}}$$ (7.14)

The volumetric flow rate written in terms of pressure gradient becomes,

$$Q = { {\pi D^4 \Delta p} \over {128 \mu l}}$$ (7.15)

(c) Aerospace, Mechanical & Mechatronic Engg. 2005
University of Sydney