Confused about DOE and repeatability

Question

I'm trying to design an experiment. I want to vary Pressure and temperature at two levels (high and low) and measure the effect on quality and flowrate.

I'm new to DOE and from my research I need to do a 2x2 full factorial design giving a total of 4 tests to run.

The literature mentions replication sets (where I do all 4 tests again) and center points (where i do 4 tests between the original tests). But how do I know how many replication sets I need to do? and whether center points are necessary.

I want to be able to report my repeatability but I thought this requires test being repeated at least 3 times to determine the variance etc.

Am I correct in saying that an ANOVA test is what will 'quantify' the effect of pressure/temperature on the quality and flowrate? and is a 2-way ANOVA the correct choice?

----- EDIT ----

1. This is a laboratory experiment and the effect of environment and other variables is minimal.
2. I have randomised the order in which my runs are done to try minimise the effect of random error and noise on the experiments
3. each experiment takes roughly 5 hours, I can do one per day
4. I have planned to do center points as a check for linearity.
5. my ultimate goal is to determine the effect that temperature and pressure have on the quality and flowrate and then suggest an operating point (Temp/pressure combination) which is the best.
6. thus far I have only completed the 2x2 base case tests.
7. Ive noticed that both temperature and pressure have significant effects.
8. I repeated a single one of those tests and the results were very similar

Answer 1

First: Is this a lab experiment or an experiment on a production plant? Especially in the last case there would probably be other variables than pressure and temperature that would influence the outcome. If so, can you control those other factors, or not? If they can be controlled, they could be included in the factorial design. If not, they could be seen as noise variables, and for instance if they are varying slowly, you could block the experiment, maybe using time intervals as blocks.

So, how many experimental runs can you do in a day?

If you replicate a $2\times 2$ factorial design only once (so $n=4$) and use a linear model including interaction for analysis $y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 +\beta_{12} x_1 x_2 + \epsilon$, you have 4 parameters to estimate so no degrees of freedom for estimation of variance. So you should replicate the design at least twice. Then, center points is usually included as an expedient way to get some check on linearity. To say more about how many times you should replicate, we need to know about your experimental variance, do you have some idea? If not, you can do a preliminary experiment just to get some variance estimate.

So now, I have made you many questions, if you can answer them maybe we can propose something more concrete. Especially, tell us what is the ultimate goal of your experiment. If that is process optimization, you need to see into response surface design.