I'm design my first circuit and plan on use a RC timer effectively as a battery (12v system). My question is since I plan on useing the bled off power to drive the device (relay coil) am I correct in needing to size the cap in terms of Joules so I get the proper wattage for the required time?

Basic number I've come up with is 12v, .100A (going to test this to get better number) on coil, and 4.5secs. Forgot the formula but remember comming up with about 17μF.

Since a Joule is a Watt second. I came up with needing 1.2W*4.5sec, 5.4 J. Right?

 

No. You have to find the pull out voltage of the relay and calculate the capacitance as for its voltage not to fall below the pull out voltage.

For example, your relay has a resistance of R and the pullout voltage is 10V.

10=12*exp(-t/RC)

Thus,
ln10=ln12-t/RC

Thus,

1/C=(-ln10+ln12)*R/t

Thus,

C=t/[R*ln(12/10)]

With your time t=4.5s and R=120R, C=0.2F.