A turbine which normally operates under a net head of \(300 \mathrm{~m}\) is supplied with water at \(2.5 \mathrm{~m}^{3} \cdot \mathrm{s}^{-1}\) through a pipe \(1 \mathrm{~m}\) diameter and \(1.6 \mathrm{~m}\) long for which \(f=0.005\). During a test on the turbine governor the flow to the turbine is gradually stopped over an interval of \(8 \mathrm{~s}\), the retardation of the water being proportional to \(t^{5 / 4}\), where \(t\) represents the time measured from the beginning of the shut-down. Neglecting minor losses and assuming an incompressible fluid in a rigid pipe with \(f\) independent of Reynolds number, determine the head at the turbine inlet and the velocity in the pipe at \(t=6 \mathrm{~s}\).

In fluid mechanics, velocity calculation is crucial for understanding how fluids move through pipes and channels. For our exercise, the initial velocity in the pipe is calculated using the flow rate equation: \( Q = A \times v \). Here, \( Q \) is the flow rate, \( A \) is the cross-sectional area, and \( v \) is the velocity. Given a flow rate of \(2.5 \, \text{m}^3 / \text{s} \) and a pipe diameter of \(1 \, \text{m} \), the cross-sectional area \(A \) is \( \frac{\pi}{4} \, \text{m}^2 \). Thus, the initial velocity is calculated as: \[ v = \frac{4 Q}{\pi} = \frac{4 \times 2.5}{\pi} = \frac{10}{\pi} \approx 3.18 \, \text{m/s} \].
This initial velocity serves as a reference for further calculations.

The Darcy-Weisbach equation is an empirical equation that relates the head loss (or pressure loss) due to friction in a pipe to the characteristics of the flow. The equation is given by: \[ h_f = f \frac{L}{D} \frac{v^2}{2g} \],
where:

• \( h_f \) is the head loss due to friction
• \( f \) is the Darcy friction factor
• \( L \) is the length of the pipe
• \( D \) is the diameter of the pipe
• \( v \) is the velocity of the fluid
• \( g \) is the acceleration due to gravity

In our example, these constants and variables feed into the equation to allow us to calculate the frictional head loss in the system.

Head loss refers to the loss of potential energy of the fluid as it moves through the pipe. This is primarily because of the friction between the fluid and the pipe walls. In the Darcy-Weisbach equation:
\[ h_f = f \frac{L}{D} \frac{v^2}{2g} \]
We use the given values: length of the pipe \(L = 1.6 \text{ m}\), diameter \(D = 1 \text{ m}\), friction factor \(f = 0.005\), and calculated velocity \(v \approx 1.094 \text{ m/s}\). Substituting these, we get:
\[ h_f \approx 0.005 \times 1.6 \times \frac{(1.094)^2}{2 \times 9.81} \approx 0.00049 \text{ m} \].
This head loss is small but essential in understanding the overall pressure changes in the system.

Flow rate is the volume of fluid passing a point in a given time, typically expressed in \( \text{m}^3 / \text{s} \). In our problem, we start with a flow rate of \(2.5 \, \text{m}^3 / \text{s} \). This is essential for calculating the initial velocity. The formula used is:
\[ Q = A \times v \]
With a known flow rate and cross-sectional area, we found the initial velocity to be \(3.18 \, \text{m/s} \).
Understanding flow rate helps in visualizing how fast or slow fluid moves through the pipe.

Retardation is the deceleration or reduction of fluid flow rate over time. Our exercise relates retardation to time using the expression \( t^{\frac{5}{4}} \). The velocity function given was:
\[ v(t) = v_0 - k \times t^{\frac{5}{4}} \]
where \( v_0 \) is the initial velocity and \( k \) is a constant. Retardation was proportional to time raised to the power \(5/4\). At \( t = 8 \text{ s} \), the flow stopped, thus \( v(8) = 0 \). Solving for \( k \), we got:
\[ k = \frac{3.18}{8^{\frac{5}{4}}} \approx 0.216 \].
This function allows us to predict the velocity at any given moment, such as at \( t = 6 \text{ s} \), resulting in a calculated velocity of \(1.094 \, \text{m/s} \). This gradual deceleration is critical for understanding fluid mechanics in dynamic systems.