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Chapter 11: Problem 10

A dam 50 meters high is constructed on a river and is delivering \(180 \mathrm{MW}\) at some moment in time. What is the flow rate of water, in cubic meters per second, if the facility converts gravitational potential energy into electricity at \(90 \%\) efficiency?

Short Answer

Expert verified
Answer: The flow rate of water in this scenario is approximately 0.32997 cubic meters per second.

Step by step solution

01

Calculate the Gravitational Potential Energy

To find the flow rate, we will first calculate the gravitational potential energy (GPE) of the water as it falls through the dam. The GPE can be calculated using the formula: GPE = m * g * h where, m = mass of water (kg) g = acceleration due to gravity (9.81 m/s^2) h = height of the dam (50 meters) However, since we don't have the mass of water directly, we will use the power generation capacity of the dam given as 180 MW.
02

Utilize the relation between Power, Potential Energy, and Time

We have the power generation capacity P given by 180 MW (180 * 10^6 W) and the potential energy calculated above. Since power is defined as the work done per unit time, we can relate the potential energy and power as follows: P = GPE / t Where t is time in seconds. Now, using the fact that the dam's energy conversion efficiency is 90%, the energy conversion can be represented as: Total_GPE = GPE * Efficiency
03

Calculate the Mass Flow Rate of Water

Now we have the total gravitational potential energy and the relation between power and time. We can find the mass flow rate (mass per unit time) using these variables: Mass_Flow_Rate = Total_GPE / (g * h) Substituting the values for power, efficiency, g, and h: Mass_Flow_Rate = (180 * 10^6 * 0.9) / (9.81 * 50) Calculating the mass flow rate, we get: Mass_Flow_Rate ≈ 329.97 kg/s
04

Calculate the Volume Flow Rate

Now that we have the mass flow rate of water, we can convert this into volume flow rate. To do this, we need to divide the mass flow rate by the density of water (ρ), which is approximately 1000 kg/m^3: Volume_Flow_Rate = Mass_Flow_Rate / ρ Substitute the value of the mass flow rate and the density of water: Volume_Flow_Rate = 329.97 kg/s / 1000 kg/m^3 Calculating the volume flow rate, we get: Volume_Flow_Rate ≈ 0.32997 m^3/s So, the flow rate of water is approximately 0.32997 cubic meters per second.

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