No I don't think you can work this work as you have two variables but only one equation.
In your OP you say that you need P1 and DP. I think you also need to know expansion volume...
Normally to work out an expansion tank volume you start with your initial P1, work out the volume it expands due to increased temperature (hence you need to know the total volume of liquid you are dealing with), have your design pressure known and then you can figure out how much air volume is required to go from P1 to design pressure. You need to assume then how much volume you start with at P1 then add the air volume so that when the expansion volume fills the tank you end up with no more than design pressure. This is the relatively simple method - of course as pressure increases the density of the fluid changes a little bit and the volume of the pipes etc changes a little bit, but these are usually fairly small compared tot he expansion of the fluid. For expansion tanks it is better to be a little cautious.
If you know three of air volume, liquid expansion volume, P1 or P2 you can work out the fourth , but I don't think you can do this by reverse unless you assume a number for P1 or P2 or the volumes.
Sorry, but without guessing I don't think this will work.
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Also: There's usually a good reason why everyone does it that way