Question

1. A jet of water of 20mm in diameter exists a nozzle directed vertically upwards at a velocity of 10m/s. Assuming the jet retains a circular cross-section, determine the diameter (m) of the jet at a point 4.5m above the nozzle exit. Take P(rho) water = 1000kg/m3.

2. A siphon has a uniform circular bore of 75mm diameter and consists of a bent pipe with its crest 1.4m above water level and a discharge to the atmosphere at a level 2m below water level. Find the velocity of flow, the discharge and the absolute pressure at crest level if the atmospheric pressure is 98.1KN/m2 neglect loses due to friction.

3. Calculate the terminal velocity of sphere of density 6000kg/m3, diameter 0.1m falling through water of density 1000kg/m^3, and its dynamic viscosity to be 0.001kg/ms. Assume that the drag coefficient is given by cd = 0.4 (Re/10,000)0.1

Answer 1

Three fluid dynamics problems are solved using the principles of mass conservation, energy conservation, Bernoulli's equation, Torricelli's law, and Stoke's law. The calculations involve jet diameters, siphon flow characteristics, and the terminal velocity of a falling sphere.

Step-by-step explanation:

Answers to Fluid Dynamics Problems

The following solutions apply to fluid dynamics and Bernoulli's principle.

1. Jet Diameter Calculation: The conservation of mass for incompressible fluids dictates that the flow rate at the nozzle exit (Q1) equals the flow rate at the point 4.5m above (Q2). Using the equation Q1 = A1 * V1 = A2 * V2, where A is the cross-sectional area of the jet, V is the velocity, and considering the jet exits at 10 m/s with a 20 mm diameter (D1), the new diameter (D2) can be found. Calculate the area at exit A1 = π(D1/2)^2, find the velocity at 4.5m using energy conservation (Bernoulli's equation), and consequently find the new diameter D2.
2. Siphon Velocity and Pressure: For the siphon problem, the velocity of flow can be found using Torricelli's law, V = √(2gh), where h is the difference in height between water level and discharge, and g is acceleration due to gravity. The discharge Q is then calculated using Q = A * V, where A is the bore cross-sectional area. The absolute pressure at the crest is found by applying Bernoulli's equation between the water level and crest level, accounting for the atmospheric pressure.
3. Terminal Velocity of a Sphere: For a sphere falling in a fluid, apply Stoke's law and balance the forces (gravitational force, buoyant force, and drag force) to find the terminal velocity. The drag force uses the given drag coefficient equation, which is based on the Reynolds number, a function of terminal velocity. Solve iteratively or by estimating the Reynolds number term to be negligible.

Answer 2

The diameter of the jet at 4.5m above the nozzle can be calculated by using the conservation of mass and adjusting for velocity change due to gravity. The siphon's velocity of flow, discharge, and pressure at the crest can be found with Bernoulli's equation and hydrostatic principles. The sphere's terminal velocity in water can be calculated balancing drag, gravity, and buoyancy, considering given density, diameter, and viscosity values.

To determine the diameter of the jet of water at a point 4.5m above the nozzle exit, we use the principle of conservation of mass. Assuming the flow is incompressible, the volume flow rate must be constant throughout the jet's path.

This implies that the product of the cross-sectional area and the velocity at any two points along the path must be equal. Therefore, A1 * V1 = A2 * V2, where A indicates the area and V indicates the velocity.

Utilizing the fact that the area of a circle is given by π * d2/4, where d is the diameter, this relationship allows us to solve for the unknown diameter (d2) at the height of 4.5m.

However, since the velocity changes with height due to gravity, we must first utilize Bernoulli's principle to find the new velocity (V2) at the height of 4.5m.

For the second part, a siphon with a uniform circular bore and a crest 1.4m above the water level, the velocity of flow (V) can be found by applying Bernoulli's equation and Torricelli's theorem.

The discharge (Q) is the product of the flow velocity and cross-sectional area. As for the absolute pressure at the crest level, we can apply Bernoulli's equation along the siphon path considering the elevation head and the atmospheric pressure given.

The terminal velocity of a sphere falling through a fluid can be found using the balance between the drag force, gravitational force, and the buoyant force. The relevant equation incorporates the drag coefficient (Cd), the sphere's density, the fluid's density, and the dynamic viscosity.