Three identical pipes of lengh L, diameter d and friction factor f are connected in parallel between two reservoirs. What is the size of pipe of length L and of the same friction factor f equivalent to the above pipe?

Concept:

(a) Series systems – the flow rate through the entire system remains constant, the total head loss in this case is equal to the sum of the head losses in individual pipes

(b) Parallel system – head loss is the same in each pipe, and the total flow rate is the sum of the flow rates in individual pipes

Calculation:

\(h_f = \frac{{fL{V^2}}}{{2gd}} = \frac{{fL}}{{2gd}}{V^2} \)

\(\frac{{fL}}{{2gd}}{\left( {\frac{{4Q}}{{\pi {d^2}}}\;} \right)^2} = \frac{{8f{Q^2}}}{{{\pi ^2}g}}\left( {\frac{L}{{{d^5}}}} \right) \Rightarrow {h_f} \propto \frac{{{Q^2}L}}{{{d^5}}}\\\)

\(\frac{{{Q^2}{L_{eq}}}}{{d_{eq}^5}} = \frac{{{{\left( {\frac{{\rm{Q}}}{3}} \right)}^2}L}}{{{{\rm{d}}^5}}}\\\)

\(\Rightarrow \frac{1}{{{\rm{d}}_{{\rm{eq}}}^5}} = \frac{1}{{9{{\rm{d}}^5}}}\\\)

\(\Rightarrow {{\rm{d}}_{{\rm{eg}}}} = {\left( 9 \right)^{\frac{1}{5}}}{\rm{d}} = 1.55{\rm{d}}\)