Consider a Y-shaped construct of three tubes. The stem has length L1

and diameter D1. The two branches are identical and have length L2 and

diameter D2. The flow is fully developed and laminar (Hagen-Poiseuille),

with negligible junction losses. The mass flow rate in the stem (m˙ 1) is

twice the mass flow rate in one of the branches (m˙ 2). The total pressure

drop (P) along the Y-shaped construct is fixed. The total volume of the

three tubes is also fixed.

Determine the ratio D1/D2 such that the total residence time in the

Y-shaped construct is minimal. The residence time in one tube is t = L/U,

where U is the average velocity. The total residence time is t = t1 + t2.

Does the optimal ratio D1/D2 depend on L1 and L2? Comment on this

optimal D1/D2 ratio and the 21/3 value obtained by minimizing the flow

resistance of the Y construct.

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