Not very easily. Probably. That picture didn't mention any specs, and I'm tired so not inclined at the moment to look up anything. Obviously the valve power draw must not exceed the control transistor ratings.
You didn't mention software, or exactly how you're going to use it.
(TIPs are pretty ancient stuff, FETs are better.)
Run the circuit first without the valve inline, and just watch that on/off timings (from your software) and voltages from the transistor look good, and it will very likely work.
Is it an AC or DC solenoid? It matters, a lot, because the AC ones rely on their inductance to limit their current and if you connect DC to them, they will overheat and/or destroy your driver transistor. DC solenoids rely on their winding resistance to limit their current; if you feed them AC at their rated voltage, they won't actuate reliably.
If you need to drive an AC solenoid, you can't just use a simple BJT.