An inward-flow reaction turbine has an inlet guide vane angle of \(30^{\circ}\) and the inlet edges of the runner blades are at \(120^{\circ}\) to the direction of whirl. The breadth of the runner at inlet is a quarter of the diameter at inlet and there is no velocity of whirl at outlet. The overall head is \(15 \mathrm{~m}\) and the rotational speed \(104.7 \mathrm{rad} \cdot \mathrm{s}^{-1}\) (16.67 rev/s). The hydraulic and overall efficiencies may be assumed to be \(88 \%\) and \(85 \%\) respectively. Calculate the runner diameter at inlet and the power developed. (The thickness of the blades may be neglected.)

Fluid mechanics is the branch of physics concerned with the behavior of fluids (liquids and gases) and the forces on them. In the context of an inward-flow reaction turbine, understanding fluid mechanics is crucial because this helps us model how water (a fluid) flows through the turbine blades. This turbine uses water's kinetic and potential energy to generate power. Since the water flow is directed inward from the periphery to the centre, the motion and behavior of fluids through the turbine are pivotal for its efficiency and effectiveness. Concepts like continuity equation, Bernoulli's principle, and potential energy transformations are heavily applied here.

Turbine efficiency is a measure of how well the turbine converts the energy in the fluid (water, in this case) into useful mechanical energy. There are different efficiency types to consider: hydraulic efficiency and overall efficiency. Hydraulic efficiency (\( \eta_h\) determines the ratio of the mechanical energy produced by the water pressure to the input energy. For our turbine, this value is 88%. Overall efficiency (\( \eta_o\) accounts for all energy losses including those due to mechanical friction and electrical generation inefficiencies. It is 85% in this exercise. High efficiencies imply that there are minimal losses and the turbine design effectively harnesses the energy available in the fluid.

Velocity triangles are graphical representations used to understand the velocities of fluids at different points in a turbine. They help illustrate the relationship between the different velocity components like absolute velocity, relative velocity, and the whirl velocity. For our problem, we have inlet and outlet triangles, where we work with the angles of the guide vane (\( \beta_1 = 30^\circ \)) and the runner blade (\( \alpha_1 = 120^\circ \)). The key part of solving turbine problems is correctly interpreting these triangles to find relationships like the tangential component of velocity (\( \C_{w1}\)) and blade speed (\( \U = \frac{\omega D_1}{2}\)). This allows us to eventually find crucial measurements like the runner diameter.

Power calculation in turbines involves determining the useful work output from the available hydraulic energy. The power developed (\( \P\)) is derived using the formula: \( P = \eta_o \rho g Q H\). In this exercise, we use the given overall efficiency (85%), fluid density (\( \rho = 1000 \ kg/m^3\)), gravitational acceleration (\( g = 9.81 \ m/s^2\)), flow rate (\( Q = 17.045 \ m^3/s\)), and head (\( H = 15 \ m\). Substituting these values into the formula gives \( \P = 213.4 \ kW\), which is the amount of power generated by the turbine under given conditions. This illustrates the practical application of theoretical aspects in hydraulic engineering.

Hydraulic engineering focuses on the flow and conveyance of fluids, primarily water and sewage. It encompasses the design and management of systems and structures like dams, channels, and turbines. Inward-flow turbines like the one in our exercise require strong hydraulic engineering principles for design efficiencies. Understanding pressure dynamics, fluid flow characteristics, and energy conversions is essential. This ensures that the water pressure and flow are harnessed optimally to produce energy. By calculating hydraulic and overall efficiencies, designing velocity triangles and ensuring the conversion of hydraulic power to mechanical energy, hydraulic engineering plays a pivotal role in the effective functioning of reaction turbines.