When fluid flows through an abrupt expansion as indicated in Fig. P5.125, the loss in available energy across the expansion, \(\operatorname{loss}_{\mathrm{ex}}\), is often expressed as

\[ \operatorname{loss}_{\mathrm{ex}}=\left(1-\frac{A_{1}}{A_{2}}\right)^{2} \frac{V_{1}^{2}}{2} \]

where \(A_{1}=\) cross-sectional area upstream of expansion, \(A_{2}=\) cross-sectional area downstream of expansion, and \(V_{1}=\) velocity of flow upstream of expansion. Derive this relationship.

Figure P5.125

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