Video: Finding the Total Annual Energy Output of a Geothermal Power Station

Hellisheiði Power Station is a geothermal power station in Hengill, Iceland. The power station has an installed capacity of 303 MW. How much energy does the power station produce each year? Give your answer in gigawatt-hours to 2 significant figures. Use a value of 365 for the number of days in a year.

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Video Transcript

Hellisheiði Power station is a geothermal power station in Hengill, Iceland. The power station has an installed capacity of 303 megawatts. How much energy does the power station produce each year? Give your answer in gigawatt hours to two significant figures. Use a value of 365 for the number of days in a year.

Okay, so in this question, we’re dealing with what’s known as a geothermal power station. And we’ve been told that this geothermal power station in Iceland has an installed capacity of 303 megawatts. In other words, the maximum power that this geothermal power station can produce is 303 megawatts. What we’ve been asked to do is to find out the amount of energy produced by this power plant each year. In other words then, we can say that we need to find the quantity 𝐸, which is the energy produced by the power plant, in a period of time which we’ll call 𝑡 of one year. Additionally, we’ve been asked to give our answer in gigawatt hours, which means that we’re going to need to find this energy produced by the power station in gigawatt hours.

Now to fully understand this, let’s recall the relationship between power, energy, and time, the three quantities that we’ve labeled here. We can recall that power is defined as the energy transferred per unit time. Or in the case of a power station, the power produced by the power station is equal to the useful energy generated by the power station divided by the time taken for this useful energy to be generated.

So if in this case we want to calculate the energy produced by our power station, then we need to rearrange the equation. We can do this by multiplying both sides of the equation by the time 𝑡, because this way, on the right-hand side, we’ve got 𝑡 in the numerator divided by 𝑡 in the denominator. 𝑡 divided by 𝑡 is just one. And so we’re only left with 𝐸 on the right-hand side, whereas on the left, we’ve got time multiplied by power. And so we could find the energy 𝐸 by simply substituting the values in for the power 𝑃 and the time 𝑡.

However, remember that we’ve been asked to give our answer in gigawatt hours. Whereas if we currently multiply time by power or power by time on the left-hand side, after substituting these quantities into our equation, then we’ll find our energy 𝐸 in units of megawatts multiplied by years. Because once again we’re multiplying power by time.

However, because we want our answer in gigawatt hours, what we can do before substituting these quantities into our equation here is to convert the power into gigawatts and the time into hours. So let’s start by looking at the power first. Let’s recall that 303 megawatts is the same as 303 times 10 to the power of six watts. Because the prefix mega- just means 10 to the power of six. But instead we want to write this as gigawatts.

Now we should recall that a gigawatt is the same thing as 10 to the power of nine watts, because the prefix giga- means 10 to the power of nine. And so this value here we want to try and write as 10 to the power of nine. We can do so by multiplying the entire quantity by 10 to the power of three divided by 10 to the power of three.

First of all, the reason we can do this is because the numerator and denominator are exactly the same. And so this fraction, 10 to the power of three divided by 10 to the power of three, is just equal to one. Essentially, we’re multiplying the right-hand side by one, which we can do because anything multiplied by one is the same thing.

However, we’re going to be clever about this here now. Because everything on the right-hand side is just one quantity with different terms multiplied together, we will take the 10 to the power of three in the numerator and multiply it by 10 to the power of six, whereas the 10 to the power of three in the denominator we will divide into 303. In other words, what we have now is 303 divided by 10 to the power of three multiplied by 10 to the power of six multiplied by 10 to the power of three watts.

Now looking at this part here, 10 to the power of six times 10 to the power of three, that altogether is just equal to 10 to the power of nine, which is why we did all this in the first place because we want to write our power in terms of 10 to the power of nine watts or gigawatts. And then 303 divided by 10 to the power of three is simply 303 divided by 1000, because 10 to the power of three is 1000. And simplifying that simply gives us 0.303.

And so to quickly recap, what we had earlier was that the power was equal to 303 times 10 to the power of six watts. But in order to write it in terms of gigawatts, we multiplied this part of our quantity by 10 to the power of three. But to keep the overall power the same, we divided this part by 10 to the power of three. And altogether, that gave us 0.303 times 10 to the power of nine watts, which we can now write as 0.303 gigawatts, due to this conversion factor that we’ve seen already.

So now that we’ve written our power in terms of gigawatts, let’s try writing our time in terms of hours. Let’s recall that the time 𝑡, which is equal to one year, is the same thing as 365 days. Because, remember, we’ve been told in the question to use a value of 365 for the number of days in one year. But then we can recall that, in one day, there are 24 hours. And therefore, we can multiply the right-hand side of our equation by 24 hours per day.

Now the reason we can do this is the same as before. 24 hours is exactly the same thing as one day. So 24 hours divided by one day is equal to one. And so we’re just multiplying by one. But the benefit of writing the numerator in terms of hours and the denominator in terms of days is that now the unit of days in the numerator and denominator cancels. And we’re left with the time 𝑡 is equal to 365 multiplied by 24 hours, which is 8760 hours. And hence, we’ve also written the time 𝑡 in the desired units.

This means that we can now come back to our equation for the energy produced by a geothermal power station. We can say that the energy 𝐸 produced by the power station in one year is equal to the power of the power station, which is 0.303 gigawatts, multiplied by the time in hours that’s equivalent to one year, which is 8760 hours.

Now what we can do is to multiply the numerical quantities 0.303 multiplied by 8760. And we can also multiply the units together, gigawatts multiplied by hours. Because this way, we’ll find the energy 𝐸 in terms of whatever the numerical value is and in units of gigawatt hours, just like we’ve been asked to in the question.

Evaluating the right-hand side of this equation then, we find that the energy produced in one year by the geothermal power station is 2654.28 gigawatt hours. However, remember that we’ve been asked to give our answer to two significant figures. So here’s our first significant figure, and here’s the second. To understand what happens to our second significant figure, we will have to look at the next value. This number is a five, and five is greater than or equal to five, which means that our second significant figure is going to round up to a seven. And hence, we found the final answer to our question. The energy produced by the geothermal power station in one year, assuming it’s working at installed capacity, to two significant figures, is 2700 gigawatt hours.

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