How do I block for "month" in a 2^(4-1) fractional factorial design?

[#1]

I have 4 factors & plan on doing 8 runs. I'm wondering if I can block 4 of the runs for the 1 month I run those experiments, and 4 of the runs the next month that I run the experiment - basically 2 blocks of size 4. Would adding month as a blocking factor mean I now have 5 factors in my design? How would that look in a model & what would that alias structure look like? 

Somewhat related - I'd like to account for lot# of a raw material in my design, but I can't really block it - it seems most runs have one of 3 or 4 lot #'s so I can't do 2 levels of lot # - would I account for this as a covariate? How would that look in a model? 

[#2]

Basically - can I block an unreplicated 2^(4-1) design using "month" as the blocking factor. 

[#3]

Realize there are two distinctly different approaches to the use of blocking and you will get differing opinions. Much depends on the specific situation for which you have provided very little context.

Here are some of my thoughts regarding blocking:

1. You can do Randomized Complete Block Designs (RCBD) or Incomplete Blocks Designs (AKA Balanced Incomplete Block (BIB)).  

2. The model depends on how you assign the block.  

a. If you have identified the noise and are specifically confounding it with the block (e.g., what is actually changing month-to-month?), you may want to treat the block as a fixed effect.  If so, you can "create" the incomplete block by bumping the "number of factors" in your 8 treatment experiment to 5.  Choose any column and sort the design.  The first 4 runs are coded block -1 and the second are 1.  This will be resolution III design.  Y = Block+X1+X2+X3+X4+Block*X1+Block*X2 

Aliasing:

b. If you have not identified the noise (which is what appears to be your case) and therefore cannot assign the the noise to the block, the block should be treated as a random effect and used to increase the inference space and quantify experimental error. Y = X1+X2+X3+X4+Block(error)

Aliasing:

 3. For covariates, you would need a continuous value for each lot.  Some measure of the lot.  There is much to consider for this (e.g., measurement errors, is the measure for the lot consistent within lot, etc.)

[#4]

I had to think about this one for a bit, but I think you can block months by confounding two of your 2-factor interactions in the 2^(4-1) design.

You'd then leave those two 2-way interactions out of your model since each block would only have a + (or -) setting for those interactions, sort of how you cannot estimate the ABCD interaction in a 2^(4-1) design since each half fraction only consists of the + levels (for one of the half fractions) & - levels (for the other half fraction ) of ABCD. I guess the issue is you'd have to choose which of your two 2-factor interaction alias chains are less likely to be relevant though. 

So the model would be something like: Y = u + X1*A + X2*B + X3*C + X4*D + X12*(2-factor interaction 1 + X*(2-factor interaction 2)