Question

Consider a steady flow through a horizontal nozzle. The nozzle inlet area is 1 m\(^2\) and the outlet area is 0.05 m\(^2\). At the outlet, the flow discharges to atmosphere. Assuming the flow to be incompressible and frictionless, and the density of the fluid as 1 kg/m\(^3\), the gauge pressure required at the nozzle inlet to produce an outlet speed of 100 m/s is ________________ N/m\(^2\) (rounded off to nearest integer).

Hint:

Always check if the flow discharges to atmosphere — this sets the outlet gauge pressure to zero, simplifying Bernoulli’s equation.

Answer 1

Correct Answer: 4987

Continuity equation: \[ A_1 V_1 = A_2 V_2 \] \[ V_1 = \frac{A_2}{A_1} V_2 = \frac{0.05}{1} \times 100 = 5\ \text{m/s} \] Apply Bernoulli equation (horizontal flow): \[ P_1 + \frac{1}{2}\rho V_1^2 = P_2 + \frac{1}{2}\rho V_2^2 \] Outlet discharges to atmosphere ⇒ \(P_2 = 0\) gauge. Density \(\rho = 1\ \text{kg/m}^3\) \[ P_1 = \frac{1}{2} \rho (V_2^2 - V_1^2) \] \[ P_1 = \frac{1}{2} (1) (100^2 - 5^2) \] \[ P_1 = 0.5 (10000 - 25) = 0.5 \times 9975 = 4987.5 \] Thus the required gauge pressure is approximately: \[ \boxed{4987\ \text{to}\ 4988\ \text{N/m}^2} \]
Final Answer: 4987–4988 N/m\(^2\)