I would guess it's a non-standard symbol for a check valve. @NMech is correct that it looks like a diode, but that would not appear on a P&ID. A check valve is a fluid flow equivalent of a diode though. I did find a reference that shows that symbol as a check valve here.
First get the newest edition of Perry's Chemical Engineers' Handbook a.k.a The Chemical Engineer's Bible which covers at the very least the basics of pretty much everything you need.
Then look at Coulson & Richardson's Chemical Engineering series and pick up the volumes you need. Pretty much everyone has used at least Volume 6 (Design) now and then. ...
First, find the energy balance - basically mass flow x thermal capacity x delta T is the same on both sides.
Edit to add
There's an error in your energy balance: on the left hand side it's simply $\Delta T$, not the derivative. Also make sure to use the correct values for $c$ as you have two different media.
Then, find something like this for your valve: