Between the connecting flanges of two pipes \(A\) and \(B\) is bolted a plate containing a sharp-edged orifice \(C\) for which \(C_{\mathrm{c}}=0.62\). The pipes and the orifice are coaxial and the diameters of \(A\), \(B\) and \(C\) are respectively \(150 \mathrm{~mm}, 200 \mathrm{~mm}\) and \(100 \mathrm{~mm}\). Water flows from \(A\) into \(B\) at the rate of \(42.5 \mathrm{~L} \cdot \mathrm{s}^{-1}\). Neglecting shear stresses at boundaries, determine ( \(a\) ) the difference of static head between sections in \(A\) and \(B\) at which the velocity is uniform, (b) the power dissipated.

The continuity equation is a fundamental principle in fluid dynamics. It states that the mass flow rate must remain constant from one cross-section of a pipe to another if there are no leaks. Essentially, if you know the flow rate at one section of the pipe, you can use it to find the flow rates at other sections using the equation:
equation: \( Q = A_1 V_1 = A_2 V_2 \)
where \( Q \) is the flow rate, \( A \) is the cross-sectional area, and \( V \) is the velocity of the fluid. This equation helps to understand how fluids speed up or slow down when moving through pipes of varying diameters. By maintaining the product of area and velocity constant, we can predict fluid behavior effectively.
For example, if fluid flows from a wider pipe into a narrower section, its velocity must increase, and vice-versa. This principle is crucial when analyzing fluid mechanics problems, especially when dealing with complex pipeline systems. In our exercise, we use the continuity equation to calculate velocities at different points using known flow rates and cross-sectional areas.

Flow rate is the volume of fluid that passes through a cross-section of a pipe per unit time. It's usually expressed in liters per second (L/s) or cubic meters per second (m³/s). The formula for flow rate is:
equation: \( Q = A \times V \)
where \( Q \) is the flow rate, \( A \) is the cross-sectional area, and \( V \) is the velocity of the fluid. Calculating flow rate is important for designing and analyzing plumbing systems, ensuring that they can handle required volumes efficiently. In our exercise, the given flow rate was 42.5 L/s, which converts to 0.0425 m³/s. Using this value, we could then determine the fluid velocities in different sections of the pipe.

The cross-sectional area of a pipe affects how quickly a fluid can flow through it. It can be calculated using the diameter of the pipe with the formula for the area of a circle:
equation: \( A = \frac{\text{π} d^2}{4} \)
where \( d \) is the diameter. In the given exercise, the diameters of the pipes are converted from millimeters to meters (0.15 m, 0.2 m, and 0.1 m for sections A, B, and the orifice C, respectively). The cross-sectional areas are then calculated as follows:

• For pipe A: \( A_A = 0.0177 \text{ m}^2 \)
• For pipe B: \( A_B = 0.0314 \text{ m}^2 \)
• For orifice C: \( A_C = 0.00785 \text{ m}^2 \)
The variation in cross-sectional area at different points is essential for determining fluid velocity and behavior in each section.

With the cross-sectional areas known, we can calculate the velocity at different pipe sections using the flow rate. The formula is derived from the continuity equation:
equation: \( V = \frac{Q}{A} \)
For our exercise:

• Velocity in pipe A: \( V_A = \frac{0.0425}{0.0177} = 2.40 \text{ m} \times \text{s}^{-1} \)
• Velocity in pipe B: \( V_B = \frac{0.0425}{0.0314} = 1.35 \text{ m} \times \text{s}^{-1} \)
• For the orifice C, we apply the discharge coefficient \( C_C = 0.62 \) to get the velocity at the vena contracta: \( V_{\text{vena}} = \frac{0.0425}{0.62 \times 0.00785} = 8.8 \text{ m} \times \text{s}^{-1} \)

Understanding these velocities is crucial for pipeline design and for identifying any potential issues that might arise.

Power dissipation in fluid systems refers to the loss of energy, often due to frictional forces and other resistances within the pipe. The power dissipated can be calculated using the formula:
equation: \( P_{\text{dissipated}} = Q \times (P_A - P_B) \)
where \( Q \) is the flow rate, and \( P_A \) and \( P_B \) are the pressure at sections A and B, respectively. This helps in determining how much energy is being lost in the system and is crucial for efficient system design. By understanding and minimizing power dissipation, engineers can design piping systems that conserve energy and operate more efficiently.