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A sudden expansion of flow in a pipe is a unique case where a theoretical result for energy result is available.

In a sudden expansion, fluid flows from a smaller cross-sectional area $A_1$ to a larger one $A_2$. This expansion creates flow separation and turbulent mixing, leading to significant energy loss.

The analysis of a sudden expansion is based on three fundamental conservation principles:

• Mass conservation:

$$A_1{V}_1=A_3{V}_3$$

• Momentum conservation:

$$p_2{A}_2-p_3{A}_3=\rho A_3{V}_3(V_3-V_1)$$

• Energy conservation:

$$\frac{p_2}{\gamma }+\frac{V_1^2}{2g}=\frac{p_3}{\gamma }+\frac{V_3^2}{2g}+h_L$$

From the momentum conservation expression, we can find the pressure difference as:

$$p_2=p_3+\rho V_3(V_3-V_1)$$

This indicates that the pressure after expansion ($p_3$) is lower due to the velocity difference created as fluid decelerates. The pressure recovery is incomplete because of energy dissipation through turbulence.

The head loss during a sudden expansion is derived from the energy conservation equation:

$$h_L=\frac{(V_3-V_1{)}^2}{2g}=\frac{V_1^2{\left(1-\frac{V_3}{V_1}\right)}^2}{2g}$$

or, alternatively we can write

$$h_L^{\text{Minor}}=K_L\frac{V_1^2}{2g}$$

The loss coefficient for a sudden expansion is expressed purely in terms of velocity and area ratios:

$$K_L={\left(1-\frac{V_3}{V_1}\right)}^2={\left(1-\frac{A_1}{A_2}\right)}^2$$