Dimensional Analysis

Through a dimensional analysis it is possible to derive an expression for "loss of head" in a pipe flow. Assuming that pressure drop is proportional to pipe length it can be shown that

$${{\Delta p} \over {{1 \over 2} \rho V^2}} = {{32 \mu l V} \over D^2} {1 \over {{1 \over 2} \rho V^2}} = {64 \over Re} \left(l \over D \right)$$ (7.25)

Thus pressure drop now becomes,

$$\Delta p = f {l \over D} {{\rho V^2} \over 2}$$ (7.26)

$$f ={{ \Delta p \left( D \over l \right) }\over {1/2 \rho V^2}}$$ (7.27)

The non-dimensional quantity f is referred to as "Darcy's Friction Factor". For a laminar flow it is given by

$$f = {64 \over Re}$$ (7.28)

It is left for the student to show that

$$f = {{8 \tau_w} \over {\rho V^2}}$$ (7.29)

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