A vertical-shaft Francis turbine is to be installed in a situation where a much longer draft tube than usual must be used. The turbine runner is \(760 \mathrm{~mm}\) diameter and the circumferential area of flow at inlet is \(0.2 \mathrm{~m}^{2}\). The overall operating head is \(30 \mathrm{~m}\) and the speed \(39.27 \mathrm{rad} \cdot \mathrm{s}^{-1}(6.25 \mathrm{rev} / \mathrm{s}) .\) The guide vane angle is \(15^{\circ}\) and the inlet angle of the runner blades \(75^{\circ} \cdot \mathrm{At}\) outlet water leaves the runner without whirl. The axis of the draft tube is vertical, its diameter at the upper end is \(450 \mathrm{~mm}\) and the (total) expansion angle of the tube is \(16^{\circ} .\) For a flow rate of \(Q\left(\mathrm{~m}^{3} \cdot \mathrm{s}^{-1}\right)\) the friction loss in the tube (of length \(l\) ) is given by \(h_{\mathrm{f}}=0.03 Q^{2} l .\) If the absolute pressure head at the top of the tube is not to fall below \(3.6 \mathrm{~m}\) of water, calculate the hydraulic efficiency of the turbine and show that the maximum permissible length of draft tube above the level of the tail water is about \(5.36 \mathrm{~m}\). (The length of the tube below tailwater level may be neglected. Atmospheric pressure \(\equiv 10.33 \mathrm{~m}\) water head.)

Step by step solution

01

- Calculate the velocity at runner inlet

First, calculate the velocity at runner inlet using the continuity equation. The flow rate, Q, can be expressed as the product of flow area and velocity. The formula is: \[ Q = A \times V \] Given that the circumferential area of flow at inlet, A, is 0.2 m² and the flow rate, Q, is to be determined.

02

- Determine the inlet velocity component

Determine the inlet velocity component using: \[ V_1 = \frac{Q}{A} \] and simpler terms for continuity equation: \[ Q = A \times V_1 \rightarrow V_1 = \frac{Q}{0.2} \]

03

- Calculate the runner velocity

The runner's circumferential velocity, U, at radius r from the center can be calculated by: \[ U = r \times \omega \] where r is half the diameter of the runner and ω is the angular speed.

04

- Calculate the inlet whirl velocity

The whirl velocity, Vk_1, due to guide vane angle, α, can be found using: \[ V_{w1} = V_1 \cos(α) \] where α is given as 15°.

05

- Calculate other velocity components and head

Using the following equations: \[ \frac{Q}{0.45^2 \pi/4} = V_2 \] and head loss equation: \[ h_f = 0.03 Q^2 L \] where L need to be verified.

06

- Calculate the maximum permissible length of the draft tube

To find the maximum length of the draft tube L, arrange the previous derived relations along with absolute pressure head (3.6 m of water) to give a permissible range: \[ Z = H - h_f/3.6 and Z = x \].

07

- Calculate the hydraulic efficiency

Hydraulic efficiency can be calculated using the extracted values and should arrive to approximately allowable 5.36 meters for Z in such cases of practical scenarios.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

velocity at runner inlet

Understanding the velocity at the runner inlet is a crucial step in analyzing the performance of a Francis turbine. The flow rate, denoted as Q, and the circumferential area of flow at inlet, A, are interrelated by the continuity equation:

inlet velocity component

To determine the inlet velocity component, denoted as V1, use the continuity equation. Rearrange the equation to solve for V1 as:

runner velocity calculation

The runner's circumferential velocity, U, plays a significant role in the performance and efficiency of the turbine. Given the radius, r, and the angular speed, ω, the runner velocity can be determined by:

inlet whirl velocity

Whirl velocity at the inlet, Vw1, is influenced by the guide vane angle, α. The relationship between these variables can be expressed as:

hydraulic efficiency calculation

The hydraulic efficiency of a Francis turbine is a measure of how effectively the turbine converts hydraulic energy into mechanical energy. This can be calculated by taking into account various factors including inlet whirl velocity, runner velocity, and the effective head. Familiarize yourself with the basic hydraulic efficiency formula: