Is it statistically inappropriate to include a regression parameter representing biological replicate?

Day D1:  performed experiment via protocol P1, obtained 3 technical replicates per condition.

Day D1:  performed experiment via protocol P2, obtained 3 technical replicates per condition.

Later on, i performed a biological replicate of these experiments...

Day D2:  performed experiment via protocol P1, obtained 3 technical replicates per condition.

Day D2:  performed experiment via protocol P2, obtained 3 technical replicates per condition. 

Right now, my design matrix is derives from the following formula:

expr  ~  protocol + condition

Is it inappropriate to use the following design matrix formula?

expr  ~  protocol  +  condition  + day

Thank you

I assume that each experiment at each day/protocol combination contains samples from multiple conditions, otherwise I don't see where the condition factor would come into play. If that's the case, blocking on day in the design matrix is fine. It's just like a paired samples analysis where you have two or more conditions per day; even if you add a day blocking factor, you can still compare between conditions within each day.

That said, you should be careful about how you handle technical replicates. Don't just throw all your technical replicates into the design matrix, as they will be treated (incorrectly) as biological replicates during variance estimation. You should really block on sets of technical replicates using duplicateCorrelation, which will consider the dependencies between technical replicates derived from the same biological replicate.