Thanks Chris
I realise that the area enclosed is zero when, as is the case for the Carnot Engine, the last of the four stages is stated as "work done on the gas"
This is the bit that i don't understand. The area enclosed is actually the area under the heat addition process BC minus the area under the heat extraction process DA(please refer to diagrams http://hyperphysics.phy-astr.gsu.edu/Hb … yc.html#c5). What I'm asking is why, for the purposes of calculating the efficiency, that the heat extraction process work is negative?

I understand that in the heat addition process, the gas does work on the piston. This is useful work. But in the heat extraction process, the work is done on the gas, but not by the piston. the piston surely doesn't need to be actively pushed back because the atmosphere does that job. If you take a cylinder at V2 and P1 and then remove heat, then the piston will be pushed back to V1 to keep P1 equal to atmospheric if P1 = atmospheric to start with.

So if you remove heat from the cylinder in this way then you can also extract useful work from the piston. ie the gas pushes the piston in the heating process which pushes the cam and then the atmosphere pushes the piston in the heat removal process, which pulls the cam.

So for the purposes of how much total work is done on the cam, why isn't the DA work added and not subtracted in the efficiency calculation?

This is almost like saying that if I push on the wall with 10 N of force, I know the wall pushes back with 10 N of force, therefore the total force is 20 N.

I think perhaps a closer annalogy would be pushing on a springy wall. I push on the wall with 10N and it deforms. While deforming, my hand moves distance d, and so work is done. If there were a plunger of a linear generator attached to my hand, with the generator body fixed in space, and the generator attached to a light bulb, then the movement of my hand with 10N for a distance d would result in the bulb glowing(ok, it's a really small bulb!)

Now after moving a distance d, my hand reaches the point where the elastic properties of the wall exert 10N back. the hand stops. But then if I cease to exert a force with my hand then the wall will push my hand back with 10N a distance d, and back to initial conditions. Again the plunger is moved and the bulb glows, be it with the reverse current. However my point is that work is done to the external system in both directions, in the case of the bulb, perhaps useful work.

I'm seeing the cylinder with enclosed gas and piston in the same way. When the pressure is higher inside then the gas does work on the piston, and yes I agree, also does work on the atmosphere. but the atmosphere at this point offers little resistance because it is at the lower pressure. So the bulk of the work can be harnessed.
In the reverse motion, the gas is the one with the lower pressure, so the atmosphere does the work and the gas does a small amount of work back in the form of resistance, but this is small compared to the work done on the piston by the atmosphere, so again it can be harnessed.

Two things strike me as important. if you take a stand-alone cylinder with enclosed gas and piston, you can get it to do useful work by adding heat.
If you take the same system and remove heat, you also can do useful work.
And ok, the work vector is in opposite directions in the two casses, but isn't the modulus of these two vectors the "useful" work and not the addition of positive and negative work?
To me it seems like AC current. One can't say that the negative part of the cycle should be subtracted from the positive part to obtain it's useful work

I mean I do understand that in the Carnot cycle  there is a certain loss of energy, that can't be convereted to work, because by necessity some of the energy from the hot source must end up in the cold sink, and so not all can be converted to work. But because of my thinking above, I can't understand how it's as simple as just subtracting the work in the second half of the cycle, from the first half. It seems it must be more complicated than that