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The pressure drop–flow rate relation for three different pipes is known to be linear, as shown in the figure below. Find the pressure drop–flow rate relation for different configurations of the three pipes: (a) series; (b) parallel; and (c) series–parallel.

Given:

The linear characteristic of three pipe sections:

Δp_A = R_A · Q_A; Δp_B = R_B · Q_B; Δp_C = R_C · Q_C

Find:

The equivalent hydraulic resistance for the three cases:

Δp_series = R_series · Q; Δp_parallel = R_parallel · Q; Δp_{series/parallel} = R_C · Q