Fluid Dynamics: Energy & Momentum Worked Examples

Water flows through a horizontal pipe. Section 1 has $D_1=150\text{ mm}$ and $p_1=200\text{ kPa}$ gauge; Section 2 has $D_2=75.0\text{ mm}$ and $p_2=100\text{ kPa}$ gauge. Neglect losses and take kinetic-energy correction factors as one. Determine $V_1$ and discharge.

A $50.0\text{ mm}$ nozzle discharges water to atmosphere from a large reservoir whose free surface is $5.00\text{ m}$ above the nozzle center. Neglect losses. Determine jet velocity and discharge.

A pump transfers $Q=0.0500\text{ m}^3/\text{s}$ of water between large reservoirs whose free surfaces differ by $30.0\text{ m}$. Total pipeline loss is $5.00\text{ m}$ and pump efficiency is $80.0\%$. Determine pump head, water power, and shaft input power.

Water flows at $2.00\text{ m}^3/\text{s}$ from an upper reservoir to a lower reservoir $25.0\text{ m}$ below. Pipeline losses total $3.00\text{ m}$ and turbine efficiency is $85.0\%$. Determine the hydraulic head available at the turbine and output power.

A siphon crest is $6.00\text{ m}$ above the source-reservoir free surface. Water speed at the crest is $3.00\text{ m/s}$, head loss from the reservoir to the crest is $0.500\text{ m}$, and atmospheric pressure is $101.3\text{ kPa}$ absolute. Determine crest gauge and absolute pressure.

A Pitot-static tube in air measures stagnation-minus-static pressure $\Delta p=3.50\text{ kPa}$. Air density is $1.20\text{ kg/m}^3$. Assuming incompressible behavior and calibration coefficient $C=1.00$, determine local speed.

A horizontal nozzle contracts from $D_1=100\text{ mm}$ to $D_2=50.0\text{ mm}$. Water discharge is $0.0200\text{ m}^3/\text{s}$, inlet gauge pressure is $200\text{ kPa}$, and the outlet jet is atmospheric. Neglect weight and wall shear. Determine the force of water on the nozzle.

A horizontal $150\text{ mm}$ elbow turns water from the positive $x$ direction to the positive $y$ direction. $Q=0.0500\text{ m}^3/\text{s}$, $p_1=150\text{ kPa}$ gauge, and $p_2=120\text{ kPa}$ gauge. Diameter is constant and weight is neglected. Determine the force of water on the elbow.

A water jet of area $A=0.00200\text{ m}^2$ and speed $V=20.0\text{ m/s}$ strikes a single flat plate moving away in the jet direction. Neglect gravity and splash momentum in the jet direction. Determine the plate speed for maximum power and the corresponding force and power.

An ideal Pelton runner continuously receives a water jet with $Q=0.0500\text{ m}^3/\text{s}$ and $V=30.0\text{ m/s}$. Assume buckets reverse the relative velocity by $180^\circ$ without friction and the runner speed is $u=15.0\text{ m/s}$. Determine tangential force and power.

A horizontal Venturi meter carries water with $D_1=300\text{ mm}$, $D_2=150\text{ mm}$, discharge $Q=0.150\text{ m}^3/\text{s}$, and $C_d=0.980$. Determine the inlet-to-throat piezometric-head difference represented by the standard calibrated relation.

Water moves locally at $V=6.00\text{ m/s}$ where static pressure is $120\text{ kPa}$ gauge. At the same elevation, estimate stagnation pressure under steady incompressible, negligible-loss deceleration.