(III) In an engine that approximates the Otto cycle (Fig. 20–7...

Question

(III) In an engine that approximates the Otto cycle (Fig. 20–7), gasoline vapor must be ignited at the end of the cylinder’s adiabatic compression by the spark from a spark plug. The ignition temperature of 87-octane gasoline vapor is about 430°C. Assuming that the working gas is diatomic and enters the cylinder at 25°C, determine the maximum compression ratio of the engine.
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Accepted Answer

Hey, everyone in this problem, we're told that in a high performance motorcycle engine, air mixed with fuel is compressed idiotically in a cylinder before ignition, the engine operates on a process that closely follows the auto cycle. The air fuel mixture which behaves as an gas enters the cylinder at a temperature of 30 °C. Due to the ignition, the temperature subsequently reaches 600 °C. We're told to assume the air fuel mixture is diatomic and given that the specific heat ratio gamma for the gas is 1.4 we're asked to calculate the maximum compression ratio of the motorcycle engine. We're given a diagram of the process here. So we have four points 1234, just showing those processes that were described in problem. We have four answer choices, option A 8.6 option B 11, option C 14 and option D 16. Now, in this case, we're looking for the compression ratio. So we, we think about going from the larger volume which is gonna be V one or V four, we'll call it V one to the smaller volume which will be at 0.2 or three. So V two or V three, we'll call it V two. You can think that the compression ratio that we're looking for is going to be equal two V one divided by V two. All right. So we want to figure out this ratio V one divided by V two. And what we want to think about is an equation that incorporates both of these trends. I mean, how can we get an equation that incorporates both of these terms? Now, what's important here is that we're told that this is an adiabatic process and because it's an antibiotic process with an ideal gas, we can recall the following relationship we have that the temperature T one multiplied by the volume V one to the exponent gamma minus one is equal to the temperature T two multiplied by the volume T two to the exponent gamma minus one. I know this looks a little bit messy in terms of what we're trying to find. But if we divide both sides by V two to the exponent gamma minus one and we divide both sides by T one, we're gonna get to start seeing a picture of what we want. OK. So when we do that, we get V one divided by V two all to the exponent gamma minus one is equal to T two divided by T one. Now, on the left hand side, we have almost exactly what we want. We have our compression ratio V one divided by V two, we just have it raised to this exponent of gamma minus one. So in order to get rid of that exponent, OK, we're gonna take an exponent of one divided by gamma minus one. So on the left hand side, we're gonna get our compression ratio V one divided by V two. And that value that we wanted on the right hand side, we are gonna have T two divided by T one all to the exponent one divided by gamma minus one. Yeah, the value on the left is what we're looking for. Everything on the right. We know we know the temperature T one, we know the temperature T two and we know the heat ratio gamma. So we can substitute in on the right hand side and solve for this ratio. Now, the only thing we have to be careful here is that we're given our temperature in degrees Celsius. In our equation, we need that temperature to be in Kelvin. So what we're gonna do is take those temperatures and add 273 to each of them in order to convert to Kelvin. And so our equation is going to be V one divided by V two. That compression ratio will be equal to. Now the temperature T 2 600 °C will be equal to 873 Calvin when we add 273 to it, that's gonna be divided by the temperature T 130 °C. When we add 273 we get 303. Kelvin, all of this is gonna be to the exponent one divided by 1.4 minus one. We can work this out on our calculator. And what we're gonna find is that this compression ratio V one divided by V two is going to be equal to about 14.0906. And that is the value we are looking for. All right, so going from V one to V two were compressing by about 14 times. If we compare this to our answer choices, we're going round to two significant digits. We can see that the correct answer corresponds with option C 14. Thanks everyone for watching. I hope this video see you in the next one.

Key Concepts

Otto Cycle

Adiabatic Process

Compression Ratio

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