An inward flow radial turbine (see Fig. P5.136) involves a nozzle angle, \(a_{1},\) of \(60^{\circ}\) and an inlet rotor tip speed, \(U_{1},\) of 30 ft \(/ \mathrm{s}\). The ratio of rotor inlet to outlet diameters is \(2.0 .\) The radial component of velocity remains constant at 20 ft/s through the rotor, and the flow leaving the rotor at section (2) is without angular momentum. If the flowing fluid is water and the stagnation pressure drop across the rotor is 16 psi, determine the loss of available energy across the rotor and the hydraulic efficiency involved.

Firstly, we are told that the flow leaving the rotor at section (2) is without angular momentum. When the flow leaves the rotor without angular momentum, the outlet velocity (or tangential velocity component at the rotor exit), denoted as \(W_{2}\), is zero. Thus, \(W_{2}=0\).

Next, the total velocity at the inlet is given by the vector sum of the radial and tangent components of velocity, we know the radial component of velocity remains constant at 20 ft/s throughout the rotor. So, the total inlet velocity can be evaluated by the formula \(C_{1} = \sqrt{V_{1}^{2}+U_{1}^{2}}\) where \(V_{1}\) is the radial velocity and \(U_{1}\) is the rotor tip speed at inlet (30 ft/s).

To compute the loss of available energy, we need to perform some fluid dynamics calculations. Firstly, we recall Bernoulli's equation, which relates the pressure, velocity and height at two points in a flowing fluid. However, as this is a radial turbine, without details implying otherwise, we can assume the net height difference is zero (i.e., z = 0). So, the form of Bernoulli's equation simplifies to \(p_{1} + 0.5*rho*C_{1}^{2} = p_{2} + 0.5*rho*C_{2}^{2}\). The loss of available energy is obtained by subtracting the useful energy change between inlet and outlet from the total energy change between inlet and outlet. Begin by using Bernoulli's equation and the equation for Kinetic energy to find the total energy at the inlet and outlet of the rotor (\(K_{1}\), \(K_{2}\)). The loss of energy can then be found as \(\Delta K = K_{1}-K_{2}\). Note that all pressure units must be consistent in Bernoulli's equation.

To determine the hydraulic efficiency we need to use the defined quantity as the ratio of the useful work done to the energy supplied between inlet and outlet. It can be formulated as \(\eta_{h} = (1-\Delta K/K_{1})*100\% \), where \(K_{1}\) is the total kinetic energy at the inlet and \(\Delta K\) is the loss of available energy. Multiply by 100% to express the efficiency as a percentage.