Your question seems to answer itself in some respects:
'...a power transmission of 13.1dBW (20 watts)...'
But an antenna doesn't really draw power, it gets fed power (energy) from a source and radiates it into space.
I think the question you really are asking is:
"How much power will my amplifier need to power my 20 $W_{RF}$ communications system?"
I had to do a similar exercise a year ago and I was confused by a lot of the terminology but I think I have the following figured out (please correct me if I'm wrong):
- The transmit sequence goes like this:
- Transmitter/transponder/transceiver generates the data signal
- Radio Frequency (RF) amplifier amplifies the signal and sends to antenna
- the antenna radiates that energy (hopefully towards something)
- RF power is different from electrical power (in terms of resource allocation)
- Amplifiers can't make power from nothing and are not 100% efficient
I know the last point seems obvious but in a sea of Watts and dBs, dBms in textbooks and specification sheets its easy to get lost (it was for me).
So you are now looking for an X-band power amplifier (or a transponder with built in amplifier, but at 20 $W_{RF}$ I think you're exclusively in the range of external amplifiers) that will output 20 $W_{RF}$. From a product's specification sheet you can find the input power required in Watts. Then use this power and the time needed to transmit to find how many W-hrs (capacity, Watts x hours) your battery needs. If you don't want to reference a specific product I would assume an amplifier efficiency ($P_{out,RF}/P_{in}$) of about 25% (based on first link below)
Some helpful links (I hope):
