Temperature of Solid State Relay
Let's do some maths:

Consider this 40 A rated triac - BTA40. 

Vto = Threshold voltage = 0.85V = Voltage drop across triac.

Rd = Dynamic resistance = 10 mohm

IT(RMS) = Current through the triac = 25 A

Power dissipation = [0.9Vto X IT(RMS)] + [Rd X {IT(RMS)} X {IT(RMS)}]

Where 0.9 = 2*sqrt(2) / pi

Putting the values, you get P = 25.375 Watts

This much power can't be dissipated by a small triac. You will definitely need a beefy heat sink (maybe a cooling fan too).

In my humble opinion, if you plan to use a triac without a heat sink and in closed spaces, you should limit yourself to 1A or so (for a continuous draw, it gives P = 0.775 watts). For the application you have mentioned, I'd suggest you to go for an electro-mechanical relay which will be cheaper and smaller as compared to triac + heat sink assembly.