Question

The intake to a hydraulic turbine installed in a flood control dam is located at an elevation of \(10 \mathrm{~m}\) above the turbine exit. Water enters at \(20^{\circ} \mathrm{C}\) with negligible velocity and exits from the turbine at \(10 \mathrm{~m} / \mathrm{s}\). The water passes through the turbine with no significant changes in temperature or pressure between the inlet and exit, and heat transfer is negligible. The acceleration of gravity is constant at \(g=9.81 \mathrm{~m} / \mathrm{s}^{2}\). If the power output at steady state is \(500 \mathrm{~kW}\), what is the mass flow rate of water, in \(\mathrm{kg} / \mathrm{s}\) ?

Short answer

The mass flow rate of water is approximately 10395.01 kg/s.

Step-by-step solution

- Identify known values and required outputs

Identify the known values:- Elevation difference, \(\triangle z = 10 \ \mathrm{m}\)- Inlet velocity, \(v_1 = 0 \ \mathrm{m/s}\)- Exit velocity, \(v_2 = 10 \ \mathrm{m/s}\)- Acceleration due to gravity, \ g = 9.81 \ \mathrm{m/s^2}\- Power output, \ P = 500 \ \mathrm{kW} = 500 \times 10^3 \ \mathrm{W}\The required output is the mass flow rate of water, \dot{m} \ (\mathrm{kg/s})\

- Apply Bernoulli's equation

Apply Bernoulli's equation between the inlet (1) and exit (2) of the turbine:\[ \frac{v_1^2}{2} + gz_1 = \frac{v_2^2}{2} + gz_2 + \frac{P}{\dot{m}} \]Here, \ z_1 = 10 \ \mathrm{m}\ and \ z_2 = 0 \ \mathrm{m}\. Also, \ v_1 = 0 \

- Simplify Bernoulli's equation

Substitute \ v_1 = 0 \ and \ z_1, z_2\ values into the Bernoulli equation:\[ g z_1 = \frac{v_2^2}{2} + \frac{P}{\dot{m}} + g z_2 \]This simplifies to\[ 9.81 \cdot 10 = \frac{10^2}{2} + \frac{P}{\dot{m}} \]Rearrange to solve for \ \dot{m} :\

- Solve for mass flow rate

Plug in the known values to find the mass flow rate \( \dot{m} \):\[ 98.1 = 50 + \frac{500000}{\dot{m}} \]Subtract 50 from both sides:\[ 48.1 = \frac{500000}{\dot{m}} \]Rearrange to find:\[ \dot{m} = \frac{500000}{48.1}\]Evaluating this gives us:\[ \dot{m} \approx 10395.01 \ \mathrm{kg/s} \]

Key concepts

Bernoulli's equation

Bernoulli's equation is a key principle in fluid dynamics that describes the conservation of energy in fluid flow. It relates the velocity, pressure, and height of a fluid at different points along its path. In our exercise, Bernoulli's equation helps us understand the relationship between the turbine's inlet and exit points.
The equation is given by: \[ \frac{v_1^2}{2} + gz_1 = \frac{v_2^2}{2} + gz_2 + \frac{P}{\frac{m}{dt}} \]
This means:

• The kinetic energy of the fluid (related to its velocity)
• Potential energy (related to its elevation)
• The power output of the turbine

Using this equation, we can analyze the changes in energy to find the mass flow rate of water flowing through the turbine.

Mass flow rate

Mass flow rate is the amount of mass passing through a point in a fluid system per unit of time. It is an essential quantity in problems involving fluid dynamics. The mass flow rate (\( \frac{dm}{dt} \)) is usually measured in \( \text{kg/s} \).
In the context of our problem, knowing the mass flow rate helps us understand how much water is being used by the turbine.
From the rearranged Bernoulli's equation, we get:
\[ \frac{P}{\frac{dm}{dt}} = 98.1 - \frac{10^2}{2} \]
Solving for the mass flow rate:
\[ \frac{500000}{48.1} \]
gives us approximately \( 10395.01 \text{ kg/s} \). This means the turbine helps to move over 10,000 kilograms of water every second.

Power output

Power output is the amount of energy produced by the hydraulic turbine per unit time. It is given in watts (W) or kilowatts (kW), where 1 kW equals 1000 W.
In our exercise, the power output of the turbine is given as 500 kW, which means the turbine produces 500,000 watts of power.
The power output is used in Bernoulli's equation to determine how much energy is being added to the system. This energy contribution helps us calculate the mass flow rate of the water. \[ P = 500 \times 10^3 \text{ W} \]
Including this power in our equations is crucial for solving for the mass flow rate accurately.

Elevation difference

Elevation difference refers to the height difference between two points in a fluid system. In Bernoulli's equation, this height difference affects the potential energy of the fluid.
For our problem, the elevation difference (\( \triangle z \)) between the turbine's intake and exit is 10 meters.
This potential energy change is described by: \[ \triangle z = z_1 - z_2 \]
Substituting this into Bernoulli's equation helps us evaluate the impact of elevation, which is converted into kinetic energy at the turbine's exit.
For example, the term \( 9.81 \times 10 \text{ m} \) translates to an increase in kinetic energy in resolving the final mass flow rate.

Velocity

Velocity is a measure of the speed at which a fluid flows in a certain direction. In a hydraulic system, the velocity of water changes from the intake to the exit.
In our scenario, the water enters the turbine with negligible velocity (\( v_1 = 0 \text{ m/s} \)) and exits with a velocity of 10 m/s (\( v_2 = 10 \text{ m/s} \)).
This change in velocity affects the fluid’s kinetic energy, which is accounted for in Bernoulli's equation:
\[ \frac{v_1^2}{2} = 0 \]
\[ \frac{v_2^2}{2} = \frac{10^2}{2} = 50 \]
This kinetic energy term is crucial for balancing the energy equation and finding the unknown mass flow rate of the water.