\(13.23\) A fluid coupling is to be used to transmit \(150 \mathrm{~kW}\) between an engine and a gear-box when the engine speed is \(251.3 \mathrm{rad} \cdot \mathrm{s}^{-1}\) (40 rev/s). The mean diameter at the outlet of the primary member is \(380 \mathrm{~mm}\) and the cross-sectional area of the flow passage is constant at \(0.026 \mathrm{~m}^{2}\). The relative density of the oil is \(0.85\) and the efficiency of the coupling \(96.5 \%\). Assuming that the shock losses under steady conditions are negligible and that the friction loss round the fluid circuit is four times the mean velocity head, calculate the mean diameter at inlet to the primary member.

Power transmission in a fluid coupling involves transferring energy from an engine to a gear-box through a fluid medium. This energy transfer is essential for many mechanical systems. The power input for the fluid coupling in our example is given as 150 kW. This power is provided by the engine and must be efficiently transmitted to the gear-box, ensuring minimal energy loss in the process. The coupling's efficiency plays a significant role here, affecting how well power is transmitted. Key parameters include the speed of the engine, the mean diameters at the inlet and outlet, and the efficiency of the coupling, which in this exercise is 96.5%. These values directly influence the performance and effectiveness of the power transmission mechanism.

In fluid dynamics, we study how fluids (liquids and gases) move and interact with various forces and boundaries. In our exercise, the oil flows through a specially designed flow passage between the engine and the gear-box. The flow of oil is characterized by its density, velocity, and flow rate. Here, the oil's relative density is 0.85, which is lesser than that of water. The velocity at which the oil exits the primary member, called the outlet velocity, is crucial in calculating the flow rate and subsequently the mean velocity head. Understanding these fluid properties helps us optimize the design of the fluid coupling to ensure efficient energy transfer.

Efficiency in fluid mechanics indicates how effectively a fluid coupling transmits power from the engine to the gear-box. Given an efficiency of 96.5% for the coupling in this exercise, it indicates that 96.5% of the engine's power is successfully transmitted to the gear-box, while the remaining 3.5% is lost to factors like friction and minor shocks. To calculate the coupling's efficiency, we use parameters such as the surface speeds at the outlet and inlet points (denoted as u_o and u_i) and the power transmitted (150 kW). High efficiency means lower energy losses, translating into better performance and reduced operational costs.

The mean velocity head in fluid mechanics is a measure of the kinetic energy in the flow of the fluid. It is derived from the velocity of the fluid and gravitational constant. In the given problem, the velocity head is used to calculate the energy losses due to friction. We first determine the mean velocity at the outlet, which is the flow rate divided by the cross-sectional area of the passage. For our example, this value is approximately 2.76 m/s. Using this velocity, we calculate the mean velocity head, factoring in the friction loss (stated as four times the mean velocity head). This calculation provides insight into the energy dynamics and the resultant impact on the fluid coupling's performance.

Flow rate calculations are essential for understanding how much fluid passes through a given section of the fluid coupling per unit of time. In our problem, we calculate flow rate (Q) to ensure the oil movement is consistent with the power transmission requirements. The formula applied uses the power (P), efficiency (η), oil density (ρ), and outlet velocity (u_o). The derived flow rate of approximately 0.0718 m³/s helps us understand the volume of oil flowing through the coupling, which directly impacts the energy transfer and the overall efficiency of the mechanism. These calculations are fundamental in optimizing the performance of the fluid coupling system.