Question 2.2: Multi-port Flow Junction Consider a feed pipe ( A, v1 ) bifu......

Modern Fluid Dynamics Basic Theory and Selected Applications in Macro- and Micro-Fluidics [1011841]

Multi-port Flow Junction

Consider a feed pipe (A, $\rm v_1$ ) bifurcating into two outlet pipes ( $\rm A_2 , v_2$ and $\rm A_3 , v_3$ ) where a small hole ( $\rm A_4$ ) has been detected in the junction area. Develop an equation for the leak $\rm Q_4$ .

$\rm\Sigma Q_{in}=\Sigma Q_{out}$ (2.7)

Approach	Assumptions	Sketch
• Reduced mass RTT, i.e., direct use of Eq. (2.7)	• Steady incompressibl e flow
• Sign convention (Fig. 2.2b)	• Fixed, nondeformable C.∀.
• Sign convention (Fig. 2.2b)	• Constant velocities

fig. 2.2b

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The fact that the inlet/outlet velocities are all constant, simplifies Eqs. (2.4) to (2.7), i.e.,

$\rm\iint\limits_{C.S.} \vec{v} \cdot d\vec{A} =0$ (2.4)

$\rm {Q}=\int_{{A}}{\vec{{v}}}\cdot\mathrm{d}{\vec{{A}}}=\int_{{A}}{v}_{\mathrm{n}}\mathrm{d}{A}={\overline{{{v}}}}{A}$ (2.5)

$\rm \dot{m}=ρ\bar{v} A=ρQ$ = constant (2.6)

$\rm −Q_1 −Q_2 + Q_3 + Q_4 = 0$ where $\rm Q_i = v_i A_i$

Thus,

\rm Q_{leak} ≡ Q_4 = v_1A_1 + v_2A_2 − v_3A_3

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