Question

A hydraulic turbine has \(85 \mathrm{m}\) of elevation difference available at a flow rate of \(0.25 \mathrm{m}^{3} / \mathrm{s}\), and its overall turbine- generator efficiency is 91 percent. Determine the electric power output of this turbine.

Short answer

Answer: To find the electric power output, first, calculate the mechanical power (P_mech) using the formula P_mech = ρghQ with the given values, where ρ is the density of water (1000 kg/m³), g is the acceleration due to gravity (9.81 m/s²), h is the elevation difference (85 m), and Q is the flow rate (0.25 m³/s). Then, calculate the electric power output (P_elec) using the formula P_elec = ηP_mech, where η is the overall efficiency (0.91). The resulting electric power output will be in Watts (W).

Step-by-step solution

Calculate the mechanical power of the turbine

First, we need to calculate the mechanical power of the hydraulic turbine using the elevation difference (h), flow rate (Q), and the acceleration due to gravity (g). The formula for the mechanical power (P_mech) is given by: $$ P_\text{mech} = \rho g h Q $$ Here, \(\rho\) is the density of water, which we will approximate to be \(1000 \mathrm{kg/m^3}\), and \(g\) is the acceleration due to gravity, which we will approximate to be \(9.81 \mathrm{m/s^2}\). We are given \(h = 85 \mathrm{m}\) and \(Q = 0.25 \mathrm{m^3/s}\). Now, plug these values into the formula and solve for the mechanical power.

Calculate the electric power output of the turbine

Next, we need to calculate the electric power output of the turbine-generator system using the overall efficiency (η). The formula for the electric power output (P_elec) is given by: $$ P_\text{elec} = \eta P_\text{mech} $$ We are given η = 0.91. Using the mechanical power calculated in Step 1, plug the values into the formula and solve for the electric power output.

Final Answer

After calculating the electric power output in Step 2, we can write down our final answer. The electric power output of the hydraulic turbine-generator system is: $$ P_\text{elec} = \text{value from Step 2} \ \mathrm{W} $$