In a vertical-shaft inward-flow reaction turbine the sum of the pressure and kinetic heads at entrance to the spiral casing is \(120 \mathrm{~m}\) and the vertical distance between this section and the tail-race level is \(3 \mathrm{~m}\). The peripheral velocity of the runner at entry is \(30 \mathrm{~m} \cdot \mathrm{s}^{-1}\), the radial velocity of the water is constant at \(9 \mathrm{~m} \cdot \mathrm{s}^{-1}\) and discharge from the runner is without whirl. The estimated hydraulic losses are: (1) between turbine entrance and exit from the guide vanes, \(4.8 \mathrm{~m}\) (2) in the runner, \(8.8 \mathrm{~m},(3)\) in the draft tube, \(790 \mathrm{~mm}\), (4) kinetic head rejected to the tail race, \(460 \mathrm{~mm}\). Calculate the guide vane angle and the runner blade angle at inlet and the pressure heads at entry to and exit from the runner

In a vertical-shaft inward-flow reaction turbine, the guide vane angle is essential for directing the water flow effectively into the runner blades. The calculation is based on the relationships among peripheral velocity, radial velocity, and the tangential velocity component. To find the guide vane angle (\beta), we rely on the equation that links these velocities. Specifically, we use the equation \ \( \tan (\beta) = \frac{v_w}{v_u} \ \), where \ \(v_w\) is the tangential velocity component and \ \(v_u\) is the peripheral velocity. Given a radial velocity (\ \(v_r\)) of 9 m/s and a peripheral velocity (\ \(u_1\)) of 30 m/s, we calculate:
\ \[ \beta = \tan^{-1}(\frac{v_r}{u_1}) = \tan^{-1}(\frac{9}{30}) = \tan^{-1}(0.3) \approx 16.7^\text{o}\ \]
Thus, the guide vane angle is approximately 16.7 degrees. This angle ensures that the water enters the runner blades with minimal velocity loss, improving turbine efficiency.

The runner blade angle at the inlet (\theta) is crucial for ensuring that the water smoothly transitions through the runner without creating whirl at the exit. Considering that the discharge from the runner is without whirl (i.e., \ \(v_{w2} = 0\)), we use the relationship between the radial velocity and peripheral velocity similar to the guide vane angle calculation.
We use the equation \ \( \tan (\theta) = \frac{v_r}{u_1} \), where \ \(v_r\) is the radial velocity and \ \(u_1\) is the peripheral velocity. Both are provided as 9 m/s and 30 m/s respectively.
Calculating the runner blade angle:
\ \[ \theta = \tan^{-1}(\frac{9}{30}) \approx 16.7^\text{o}\ \]
The runner blade angle at the inlet therefore is also approximately 16.7 degrees, ensuring efficient flow of water through the runner and reducing hydraulic losses.

Pressure head is a measurement of the potential energy of water and is integral in calculating the performance of a turbine. It considers the height of water and its velocity. The pressure head at different points in the turbine can be computed using Bernoulli's principle, where the total head (\text{H}) is the sum of kinetic and potential head.
Given:
\ \[ H_R = 103.4 \text{ m}, \ H_e = 102.61 \text{ m}\
\]
For accuracy, the relationship includes heights and velocities, adjusted for any losses:
\ \[ H_{total} = \text{kinetic head + pressure head}\ \]
Total energy at different points, factoring in kinetic energy and velocity, helps us derive these heads precisely. For example, at the runner entry (\text{H_R}) ,
\ \[ H_{runner} = H_V - H_{loss 2} = 112.2 - 8.8 = 103.4 \text{ m}\ \]
With kinetic energy and velocity consideration, this approach is systematic for turbine analysis.

Hydraulic losses are the energy losses occurring due to friction and other resistances within the turbine components. Understanding and calculating these losses can significantly improve turbine efficiency. In the given problem, the losses are provided at different sections of the turbine.
The hydraulic losses include:
\ \1. Entrance to exit of guide vanes: \ \text{4.8 m}\
2. Losses in runner: \ \text{8.8 m}\
3. Draft tube losses: \ \text{0.79 m}\
4. Kinetic head rejected to tail race: \ \text{0.46 m}
\ \ Partial sum gives high precision required for understanding turbine operations effectively. Each of these losses can be summed to understand the total energy lost within the turbine. For example:
\ \[H_{total loss} = H_{loss 1} + H_{loss 2} + H_{loss 3} + H_{loss 4} = 4.8 + 8.8 + 0.79 + 0.46 = 14.85 \text{ m}\ \]
Each section's loss, measured in meters, reflects both kinetic and potential heads, ultimately impacting total head and efficiency calculations. Understanding these help in optimizing turbine designs and minimizing unwanted energy dissipation.