(II) Energy may be stored by pumping water to a high reservoir...

Question

(II) Energy may be stored by pumping water to a high reservoir when demand is low and then releasing it to drive turbines (Fig. 20–15) during peak demand. Suppose water is pumped to a lake 105 m above the turbines at a rate of 1.00 x 10⁵ kg/s for 10.0 h at night. How much energy (kWh) is needed to do this each night?

Accepted Answer

Everyone in this problem. We're asked to suppose that there's a system where water is pumped to a rooftop tank, 80 m above ground level, the water is pumped at a steady rate of 1.2 multiplied by 10 to the exponent 5 kg per second or 12 hours. And we're asked to determine the total energy consumption in kilowatt hours needed to pump water to this rooftop tank. We have four answer choices again, all in kilowatt hours, we have option A 1.13 multiplied by 10 to the exponent six. Option B 7.85 multiplied by 10 to the exponent six. Option C 94.1 multiplied by 10 to the exponent six. And option D 7.85 multiplied by 10 to the exponent nine. So what we're gonna do here is we're gonna assume that there are no dissipative forces. If there's no dissipative forces, then all we need to do in order to get this water pumped up to the rooftop is to overcome the gravitational potential energy. Ok. So the total energy consumption that we have is just going to be the energy required to overcome that gravitational potential energy. OK. So it's gonna be equal to the gravitational potential energy. So our energy consumption we're just gonna call it econs is just gonna be equal to the gravitational potential energy ug. And we know that the gravitational potential energy is given by m multiplied by G multiplied by. Now the acceleration due to gravity, we know the height we know, OK. We can assume that the ground level is at height zero or pumping 80 m above that. So the height will be 80 m. This mass is a little bit trickier in this case. And we're not given the mass of water, we wanna pump, we're given a rate and an amount of time. Ok? So we know that we can pump 1.2 multiplied by 10 to the exponent 5 kg per second. Ok? So for every second, we're pumping that much and we're gonna pump for 12 hours. So what we could do is convert our 12 hours into seconds. If we multiply that by this rate, kilograms per second multiplied by seconds, we're gonna get that mass of water that we're pumping. And we would substitute that into our equation. We have our mass acceleration due to gravity and the height. The problem with that is that when we get to the end, we're gonna have kilowatt seconds instead of kilowatt hours, we're gonna have to do another conversion. So instead of converting our 12 hours, we're actually gonna leave it alone. Ok. So what we're gonna have is we're gonna have that we pump 1.2 multiplied by 10 to the exponent 5 kg per second for 12 hours or multiplying by 12 hours. And those two terms multiplied together, give us the mass of water that we're pumping. Then we multiply by the acceleration due to gravity 9.8 m per second squared and we multiply by our height of 18 years. Now, we're gonna look specifically at the units right now. And you're gonna see exactly why we left this as an hour if we think about the kilogram. And then we think about these last two terms, we have meters per second squared multiplied by meters. So we have kilogram meters squared per second squared and recall that this is equivalent to a jewel. So we think about the units we have here, we have Joel, we still have this per second from the first term. So we have dual per second and we still have our hours. OK? So this is what we're left with D per second multiplied by hour. We know that a Jel per second is a watt. And so this leaves us with watt hours, which is exactly what we want. And we'll just have to convert from watt hours to kilowatt hours. So that's why we left that 12 hours and hours. Instead of converting into seconds, it saves us converting one way and then having to convert back All right. So when we work this all out, out on our calculator, we get 1.12 896 multiplied by 10 to the exponent nine watt hours. And again, we have to be a little bit careful because the question one kilowatt hours, we're gonna divide by 1000 to get to kilowatt hours. And we can write this as 1.12896 multiplied by 10 to the exponent six kilowatt hours. And that is that final answer we were looking for that is a total energy consumption needed to pump this water to that rooftop tank. If we compare this to the answer choices we were given, we round to three significant digits. We can see that the correct answer is going to be option a 1.13 multiplied by 10 to the exponent six kilowatt hours. Thanks everyone for watching. I hope this video helped see you in the next one.

Key Concepts

Gravitational Potential Energy

Power and Energy Conversion

Flow Rate

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