Question

Water of density 1000 kg/m³ flows at a steady rate in a converging duct of circular cross-section. Let the areas of sections 1 and 2 be 100 cm² and 50 cm² respectively. The pressure and average velocity at section 1 are 110 kPa and 1 m/s.

If the centres of the two sections are at the same elevation, then the pressure at section 2 is ______ kPa (answer in one decimal place).

Hint:

Use continuity to get the velocity at section 2, then apply Bernoulli's equation between the two sections.

Answer 1

Correct Answer: 108.5

Step 1: Find the velocity at section 2 using continuity.
For steady incompressible flow, \(A_1 V_1 = A_2 V_2\).
With \(A_1 = 100\ \text{cm}^2\), \(A_2 = 50\ \text{cm}^2\), \(V_1 = 1\ \text{m/s}\): \[ V_2 = V_1 \frac{A_1}{A_2} = 1 \times \frac{100}{50} = 2\ \text{m/s} \]

Step 2: Apply Bernoulli's equation between the two sections.
Since both centres are at the same elevation, the elevation terms cancel, leaving: \[ P_1 + \frac{1}{2}\rho V_1^2 = P_2 + \frac{1}{2}\rho V_2^2 \]

Step 3: Solve for \(P_2\).
\[ P_2 = P_1 + \frac{1}{2}\rho\left(V_1^2 - V_2^2\right) = 110000 + \frac{1}{2}(1000)(1^2 - 2^2) \]
\[ P_2 = 110000 + 500(1-4) = 110000 - 1500 = 108500\ \text{Pa} \]

Final Answer:
Converting to kPa, the pressure at section 2 is 108.5 kPa.
\[ \boxed{108.5} \]