91Ó°ÊÓ

The Bernoulli equation is a fundamental principle in fluid dynamics that describes the behavior of a moving fluid. It relates the pressure, velocity, and height of the fluid at different points along its flow. In this exercise, the Bernoulli equation enables us to connect the velocity at the pump inlet and the nozzle by considering energy conservation. The simplified form of the Bernoulli equation used here is:

\[ \frac{p_1}{\rho} + g z_1 + \frac{V_1^2}{2} = \frac{p_2}{\rho} + g z_2 + \frac{V_2^2}{2} \]

In this case, we know the pressures (\( p_1 \) and \( p_2 \)) at both points are equal, allowing us to cancel them out. The equation thus relates the height difference between the nozzle and the pump (\( z_2 - z_1 \)) to the kinetic energies at those points, ultimately helping to determine the fluid velocities.

Understanding cross-sectional areas is key to determining various fluid properties like velocity and flow rate. In this exercise, we calculate the areas of the inlet pipe and the nozzle using the formula for the area of a circle:

\[ A = \pi \left( \frac{d}{2} \right)^2 \]

For the nozzle, with a diameter \( d_2 = 2.5 \text{ cm} \):

\[ A_2 = \pi \left( \frac{0.025}{2} \right)^2 \approx 4.91 \times 10^{-4} \text{ m}^2 \]

For the inlet pipe, with a diameter \( d_1 = 5.0 \text{ cm} \):

\[ A_1 = \pi \left( \frac{0.05}{2} \right)^2 \approx 1.96 \times 10^{-3} \text{ m}^2 \]

These areas are crucial for determining the velocities at these points using the relationship between volumetric flow rate and velocity.

The volumetric flow rate, \( Q \), is the volume of fluid that passes through a cross-sectional area per unit time. It is a key quantity in describing fluid flow and is given by:

\[ Q = \frac{P}{\rho g \left( z_2 - z_1 \right)} \]

Given the problem's values:

\[ P = 8.6 \times 10^3 \text{ W}, \ \rho = 1000 \text{ kg/m}^3, \ g = 9.81 \text{ m/s}^2, \ \left( z_2 - z_1 \right) = 4 \text{ m} \]

The volumetric flow rate is:

\[ Q = \frac{8.6 \times 10^3}{1000 \times 9.81 \times 4} \approx 0.2194 \text{ m}^3/\text{s} \]

This value provides the volume of water being pumped per second, essential for determining the other parameters like velocity and mass flow rate.

Pump power input is the energy supplied to the pump to move the fluid. It's given in watts (W) and in this problem, it's provided as \( 8.6 \text{ kW} \) or \( 8.6 \times 10^3 \text{ W} \). This value is used to calculate the volumetric flow rate using the formula:

\[ Q = \frac{P}{\rho g \left( z_2 - z_1 \right)} \]

The power input indicates how much work the pump does to move the water through the height difference \( z_2 - z_1 \). By knowing this, we can understand how effective the pump is in generating the necessary flow rate, which directly influences the velocities and eventual mass flow rate.

Fluid dynamics involves the study of fluids (liquids and gases) and the forces acting on them. In this problem, fluid dynamics principles help us understand how the pump pushes water through the hose and nozzle. Key concepts include:

• Velocity: Speed of water at both the inlet and outlet.
• Pressure: Constant in this exercise but crucial in many fluid flow problems.
• Flow rate: Combines velocity and cross-sectional area.
• Energy conservation: Basis of Bernoulli's equation.

Incorporating these ideas allows us to apply the Bernoulli equation and calculate velocities using the known pump power, elevation difference, and areas. This comprehensive approach ensures accurate calculations and a deeper understanding of fluid behavior.