 The next gas power cycle that we want to take a look at is that of the internal combustion engine but with compression ignition, not the spark ignition process that we looked at for the auto but this is the compression ignition. This is referred to as being the diesel cycle and we will look at an ideal version of the diesel. So the diesel is sometimes referred to as being constant pressure combustion and we'll see that when we look at a PV diagram why we would call it a constant pressure combustion process. So let's take a look at our PV diagram and the diesel is similar to the auto with one exception and that is the way that we do the heat addition process or where the combustion actually occurs. So we start off and we're at bottom dead center just like we were for the auto cycle so we'll be at state one. Now what we do with the diesel is we go through and we have a much larger greater compression ratio than we did for the auto and consequently what happens is the air now we do not have fuel it's only air that is being compressed goes to a higher temperature and a higher pressure and that is when we're at top dead center and then what the diesel does is that is when you inject the fuel at that point and given that we're at such a high temperature that you will then get the combustion process taking place because of the high temperatures and so you're injecting fuel at the end and then you begin to expand so you're at top dead center you've compressed it's at a very high temperature you inject the fuel it then starts to expand and we can then refer to that as being constant pressure expansion again that's a bit of an approximation but we'll approximate it to be constant pressure expansion taking us to state three so this here is where we are doing our combustion with which if you recall we model that as a heat addition process and then we go into another isentropic process and that is an isentropic expansion just like we did in the auto and so we come back down to state four from four to one we do heat rejection which is basically exhaust however it's modeled as a cycle so really we don't dump the gases out but we will model it as heat rejection but really it's the exhaust cycle so this is isentropic compression here and this is isentropic expansion or your power stroke so that is the difference between the auto and the diesel it's mainly with what's going on here with our heat addition process so if you recall when we talked about the auto cycle we said if you have too high of a compression ratio you could have auto ignition or knock pinging in the engine here although if you listen to a diesel which i'll show in the next video it does make a lot of noise and it does sound like a bit of a pinging but here what we're doing is we're injecting the fuel after and and that is what starts the compression so or the the combustion process so we do not have the auto ignition or the the knock problem i guess you should say because we do have auto ignition we don't have knock that you would with the auto another parameter that we will use when we look at the thermal efficiency for the diesel is that of the cutoff ratio so i'll write out the cutout ratio now so the cutoff ratio remember before we had the compression ratio r now the cutoff ratio is little r subscript c and that is the volume at state three divided by the volume at state two given we're dealing with the closed system same mass we can write the specific volume of three to two so that is the cutoff ratio which we will use when we come up with an expression for the thermal efficiency of the diesel cycle so let's take a look at that now in doing this what i'm going to do is i'm going to look at the heat addition process so basically going from state two to three and i will also look at the heat rejection process going from state four to state one so what we'll do is we will express the first law for both of those processes and then we'll work through it and come up with an expression for the thermal efficiency of the diesel so for the first law what we want to look at is two to three which was our constant pressure combustion or heat addition process so writing out the first law now we have this term here boundary work when we looked at the auto cycle we were able to reject that because it was a constant volume combustion process we no longer have constant volume combustion when we're going through the heat addition we're actually increasing the volume and consequently we're doing boundary work with our fixed mass system so what we're going to do we're going to make the substitution or the the change that we saw when we looked at the first law for closed systems and that is you can encapsulate the boundary work with the internal energy and re-express it as enthalpy so let's do that and we can write q in is equal to the enthalpy change so now we're taking into account both internal energy and boundary work and we can then rewrite that if we assume constant specific heats so that is our first expression now looking at process one through four there we have q out and remember our designation was that q in is positive for this system if it's out it's minus so that becomes a negative minus the work is equal to the change in internal energy between states four and one now here it is a constant volume and consequently the boundary work term disappears and what we're what we're left with then is q out equals u four minus u one minus q out so we can take that now and we can plug it into our expression for the thermal efficiency of the diesel so we have that and now what we're going to do we have an expression for q out and we'll plug it in there and we have an expression for q in and we'll plug it in there now the thing that I should say about the q out this was u four minus u one we can substitute that as c v t four minus t one so again we'll be able to plug in an expression in terms of the temperatures we don't like that we like to have in terms of the compression ratio so we can rearrange and get to one with the compression ratio but let's take a look at what the thermal efficiency of the diesel is so the thermal efficiency of a diesel cycle now this is an ideal because we're making approximations about the specific heat not changing and if you notice so we have this part this is the exact same as the auto but now it is modified by a term in square brackets that has the cutoff ratio embedded within it ours are a compression ratio r c is the cutoff ratio and k is the ratio of specific heats so this is identical to what we had with the auto with the exception of this term here and it turns out that that term in square brackets is actually always greater than or equal to one and with that what it means is that the thermal efficiency of the diesel uh is always going to be less than that of the auto according to this expression now that might be surprising because some of you may be aware that diesels get better fuel economy than the auto so how can it be that the diesel has a lower thermal efficiency than the auto well the place where we're able to correct for this is the fact that the diesel is operating with a much higher compression ratio and consequently we can get higher thermal efficiencies for a diesel diesels tend to be somewhere in the ballpark depending upon the design of the engine in the 30 to 40 percent ballpark and consequently it is a higher thermal efficiency than we saw for the auto the other thing is given that the diesel is compression ignition you can combust or burn a lower quality fuel in a diesel and consequently theoretically it should be cheaper uh although some parts of the world diesel and and standard gasoline there really is not that much of a significant difference but in terms of upgrading and refining and everything the fuel should be a little cheaper economically uh so that's the diesel cycle what we're now going to do is take a little bit of a look at the compression ratio and i'll show you a quick video of a diesel operating and you can compare it back to the auto where you'll notice there is significant difference in the sound of the two