[christofoo]

Now that we've established that there is a plausible mechanism by which DCD can reduce pumping losses, for my next 'magic' trick, I will now demonstrate that DCD and lean-burn cycles can be thermodynamically equivalent to a first order, again neglecting dynamic analysis for the moment, consistent with the analysis we already made.

It seems silly enough now, but I - seriously - thought everyone would slap their heads and say 'of course, DCD and lean burn are thermodynamically equivalent!' after reading this post:

Originally Posted by christofoo

...
DCD vs lean-burn: Why does lean-burn work and DCD not?

Suppose the lean-burn engine is operating at 25% (er, how to say this?) leanness, by which I mean stoichiometric AFR X 1.25.

Compare to DCD with 25% of cylinders deactivated, 75% at stoichiometric AFR. All else being equal, including valve timing.

If you think I'm wrong, explain to me:

1. Is there a difference in air mass flow rate?
2. Is there a difference in manifold vacuum?
3. Whatever else you think requires consideration.

Clear as mud, right? This is what it's like to be a physicist. I often don't know what about my communication is not going to jump out at you with perfect clarity. After the discussion with t vago, I feel inspired to give this a detailed step by step analysis, since I've been shown the value of doing so. (Although note that there is a mistake in the above; I should have said that the lean burn engine should be setup with stoich-AFR X 1.333 instead of 1.25 - I got 3/4 and 4/3 turned around. This will be further illustrated.)

How can I prove thermodynamic equivalence? As a very quick overview there will be three basic steps: assume a given lean burn engine, setup a similar DCD engine with the same fuel input and air mass flow rate, show that the DCD engine has identical thermodynamic parameters.

Now in detail, using the same definitions t vago used starting in #222:

1. Take a lean-burn engine as a given, with some set lean-AFR, produced work, manifold pressure, pumping loss, and available work.
2. Define lean-AFR = stoichiometric-AFR X K, where K>1 is my definition of 'how lean' the engine is. As a specific example, we're going to consider K=4/3=~1.33, because this is going to help illustrate a simple case later on, but this analysis will hold true for all values K>1 as long as we stay within both physical limits of lean-burn and DCD (hint: they may be different).
3. Setup a DCD engine with the same air mass flow rate and fuel input, and everything else the same. At this point we will not know pumping loss, or available work since we don't know the manifold pressure, but we do know that this engine will have the same produced work as the lean-burn engine. (Careful, I'm coming back to this assumption.)
4. How do we get the air mass flow rate to match the lean-burn engine? For the lean-burn engine, we have (air mass = fuel mass X K X stoich-AFR). For DCD, we need to introduce (air mass = fuel mass X stoich-AFR / duty-cycle), where (duty-cycle = number of cylinders that fire / total number of cylinders per cycle) and duty-cycle<1. You might think that duty cycle could be only certain rational numbers, like 3/4, 1/2, or 1/4, but actually duty cycle can be any number between 0 and 1 if you average over an arbitrarily large number of revolutions instead of only one revolution. So if we set the air mass of the DCD engine to equal to the air mass of the lean-burn engine, we get (fuel mass X K X stoich-AFR = fuel mass X stoich-AFR / duty-cycle), which reduces to (K=1/duty-cycle) or (duty-cycle=1/K). For example, In the special case where K=4/3 in a 4 cylinder lean-burn engine, a corresponding DCD engine would have duty-cycle=3/4, or in other words, 3 cylinders activated and 1 cylinder deactivated.
5. The fact that duty-cycle=1/K doesn't actually matter. The point is that there always exists some similar DCD engine which can have the same air mass flow rate and fuel input as the lean-burn engine. What we know is simply that this case exists. Or For any lean-burn engine with K, there exists some DCD engine with duty-cycle=1/K that has the same fuel input and air-mass-flow rate (and all other reasonable common denominators).
6. However, we know that air-mass-flow rate is dictated in an Otto cycle by manifold pressure. We want to assume common denominators between the two engines, so they both have the same number of cylinders, the same displacement, and the same valve timing. Since the two engines have the same air-mass-flow rate, and all else is equal, then the DCD engine must have the same manifold pressure as the lean-burn engine.
7. Now we can compare thermodynamic efficiency. We assumed that the two engines have the same fuel input and therefore they have the same produced work. We assumed they have the same mass flow rate and proved that this is achievable (within limits, I will return to this). (We also assumed the same displacement and valve timing.) Now we know that they have the same manifold pressure and therefore they have the same pumping loss and the same available work.
8. Therefore, a DCD engine having duty-cycle=1/K will have the same thermodynamic efficiency as a lean-burn engine with lean-AFR=stoichiometric-AFR X K.

Next I'm going to dive right into interesting implications that this proof has, and then after that I'll address the validity of key assumptions, which may or may not dovetail into some speculation about practicality of DCD.