Volumetric flow rate and average velocity in a pipe

10 Volumetric flow rate and average velocity in a pipe 

The volumetric flow rate is determined by writing the equation for the volumetric flow rate across an infinitesimal element of the flow area then integrating the equation over the whole flow area, ie the cross-sectional area of the pipe. It is necessary to use an infinitesimal element of the flow area because the velocity varies over the cross section. Over the infinitesimal area, the velocity may be taken as uniform, and the variation with r is accommodated in the integration. 

A typical element is a strip perpendicular to the axis (and therefore perpendicular to vx) with an inner radius r and outer radius r+Sr where 8r is infinitesimally small. To the first order in 5r, the area of this element is equal to 2-nr8r. The volumetric flow rate 8Q across this area is therefore 

The total volumetric flow rate through the pipe is obtained by integrating the element equation over the whole cross section of the pipe, that is from r = 0 to r = r,  

This expression applies to any symmetric flow in a pipe. 

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