A typical coal-fired power plant burns 300 metric tons of coal every hour to generate 750 MW of electricity. 1 metric ton = 1000 kg. The density of coal is 1500 kg/m³ and its heat of combustion is 28 MJ/kg. Assume that all heat is transferred from the fuel to the boiler and that all the work done in spinning the turbine is transformed into electric energy. a. Suppose the coal is piled up in a 10 m ✕ 10 m room. How tall must the pile be to operate the plant for one day?

Video transcript

Hey everyone. So this problem is dealing with heat engines. Let's see what it's asking us. We have a natural gas power plant that generates 600 megawatts of electricity by burning natural gas at a rate of 1800 m cubed per hour. The density of natural gas is 0.75 kg per meters cubed at a standard temperature and pressure. If the natural gas is stored in a cylindrical tank with a diameter of m, how tall must the tank be to operate the plant for 24 hours? Assume there is no heat loss. And the total work done is conserved. Our multiple choice answers here are a 22 m b 550 m c 17 m or D 0.86 times 10 to the third meters. So the key to solving this problem is recognizing that it's actually a geometry problem and not a heat transfer or work and energy problem. So we're given the rate that the gas is burned. So I'm going to highlight this the important information here. So we're given the rate of burn per volume per time and the total time that we need to use the gas. So our total volume is simply going to be 1800 cubic meters multi per hour, multiplied by 24 hours. And that comes out to 43, cubic meters. Now, the problem asks for how tall the tank needs to be. And so that is the height of the tank. We know the tank is a cylinder. And so we can recall that the equation for the volume of a cylinder is pi R squared H isolating age or height or how tall the tank must be. On the left hand side of the equation, we get volume divided by pi R squared. Now, we just saw for the volume that's 4 43, m cubed divided by pi multiplied by our radius. Now, the problem doesn't give us a radius but it does give us the diameter and we know that the diameter is two times the radius. And so the radius is 25 m. So that quantity is squared, we plug that into our calculator and we get 22 m as our final answer that aligns with answer choice. A. So the tricky part to solving this problem is recognizing the information you need because they give you a few pieces of information that you don't need. All right, so that's all we have for this one. We'll see you in the next video.